Rotation Sets and Entropy A Dissertation Presented by Jaroslaw Marcin Kwapisz to The Graduate School in Partial Fulllment of the Requirements for the Degree of Doctor of Philosophy in Mathematics State University of New York at Stony Brook August 1995 State University of New York at Stony Brook The Graduate School Jaroslaw Marcin Kwapisz We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of the dissertation. John Milnor Distinguished Professor of Mathematics Dissertation Director Yair Minsky Assistant Professor of Mathematics Chairman of Defense Mikhail Lyubich Associate Professor of Mathematics Folkert Tangerman Visiting Assistant Professor Department of Applied Mathematics and Statistics Outside Member This dissertation is accepted by the Graduate School. Graduate School ii Abstract of the Dissertation Rotation Sets and Entropy by Jaroslaw Marcin Kwapisz Doctor of Philosophy in Mathematics State University of New York at Stony Brook 1995 This thesis is concerned with the dynamics generated on the two dimensional torus by a homeomorphism isotopic to the identity. For each rotation vector of such a map, we dene the topological entropy at the rotation vector ; this is a function having the rotation set of the map for its natural domain and the topological entropy of the map for its maximum. We show how the ideas of the thermodynamical formalism aid the evaluation of this function. For maps that are pseudo-Anosov (relative to a nite set), we demonstrate its strict convexity and real analyticity. We then describe those Gibbs states that realize the topological entropy at the rotation vector as their measure theoretic entropy. iii In the general case, we show that the topological entropy is positive at the rotation vector that lies in the interior of the rotation set. We prove that any such vector is, in fact, the common rotation vector for points of a compact invariant set that carries positive topological entropy. Presumably, the entropy on this set can be made arbitrarily close to the topological entropy at the corresponding rotation vector. We verify this for pseudo-Anosov maps and prove that, in general, both entropies are explicitly estimated from below in terms of the size of the rotation set and the relative distance of the vector from the boundary. A key tool here is the theory of quasi-conformal mappings. Finally, we analyze the rotation sets arising in a model of a resonantly kicked charged particle in a constant magnetic eld. This provides a physically interesting example to which our general theorems apply. iv To Agnieszka; To the wind Contents List of Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii Acknowledgements : : : : : : : : : : : : : : : : : : : : : : : : : ix Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1 An estimate of entropy for toroidal chaos : : : : : : : : : : : 11 1.1 Statement of the result : : : : : : : : : : : : : : : : : : : : : : 11 1.2 Comparison with pseudo-Anosov maps : : : : : : : : : : : : : 14 1.3 Entropy and conformal structures : : : : : : : : : : : : : : : : 16 1.4 Proof of the Main Lower Bound : : : : : : : : : : : : : : : : : 19 2 Topological entropy at the asymptotic average : : : : : : : : 28 2.1 Relative variational principle and thermodynamical formalism 29 2.2 Topological entropy at the asymptotic average for subshifts of nite type : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 2.2.1 Transfer matrix, Gibbs states, and explicit formulas : : 45 2.2.2 Observable shadowing with the right entropy : : : : : : 53 2.2.3 The scaold of subshifts of nite type : : : : : : : : : : 61 vi 3 Topological entropy at the rotation vector for torus maps 68 3.1 Denitions and the main result : : : : : : : : : : : : : : : : : 3.2 The central example: pseudo-Anosov maps : : : : : : : : : : : 69 72 3.3 Proof of the main result : : : : : : : : : : : : : : : : : : : : : 77 3.4 Example : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 81 4 Rotation sets for the resonantly kicked linear oscillator : : 87 4.1 4.2 4.3 4.4 The physical context : : : : : : : : : : : : : : : : : The mapping and its rotation set : : : : : : : : : : The recurrence, higher period orbits, monotonicity : A mechanical toy-model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 : 90 : 94 : 100 A Elementary large deviations for topological dynamics : : : 102 Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 112 vii List of Figures 3.1 3.2 3.3 3.4 When a and b are not antipodal. The map h# on the spine. : : : : The ecient map g1 . : : : : : : : The transition graph. : : : : : : : viii : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 82 83 84 Acknowledgements I am indebted a great deal to my adviser John Milnor and to Philip Boyland for their support during my four years at Stony Brook. I would also like to thank Michal Misiurewicz and Feliks Przytycki whose teaching and mentoring began my adventure with dynamical systems. I acknowledge the Alfred Sloan Foundation for supporting this work through the doctoral dissertation fellowship. I owe special thanks to Agnieszka Bielinska, Benjamin Hinkle, and Albert Fisher for many remarks that improved this text, and to Scott Sutherland and Brian Yarrington for their help with the computer typesetting. ix Introduction This work originated from an attempt to understand certain aspects of dynamics generated on a two-dimensional torus by a homeomorphism that is isotopic to the identity. Our main object of study is the interplay between the rotation set and the topological entropy. We put these two classical invariants in the same context by using the ideas of the thermodynamical formalism, and we nd a qualitative relation between them by invoking the methods of the theory of quasi-conformal mappings. In what follows, we introduce the basic concepts, advertise our main results, and show how they are a continuation of the work by others. We tend to be brief since a signicant amount of expository material can be found in the subsequent chapters. Our favorite model of a two-dimensional torus T2 is that obtained by identifying points of R2 which dier by a vector having integer coordinates. The resulting projection : R2 ! T2 is a universal covering and we x it once and for all. The space of all homeomorphisms of T2 that are isotopic to the identity will be denoted by H(T2 ). If f : T2 ! T2 is in H(T2) then it lifts to F : R2 ! R2 which commutes with the deck group, i.e. F (~x + v) = F (~x) + v 1 2 for all x~ 2 R2; v 2 Z2 ; we will write H~ (T2) for the space of all lifts of elements of H(T2 ). Note that a map F 2 H~ (T2) is a C 0-bounded perturbation of the identity by ~F = F id : R2 ! R2 which, being invariant under precompositions with the deck translations, descends to a continuous function F : T2 ! R2 that is called the displacement of F . Thus, for x~ 2 R2 and x = (~x), F n(~x) x~ = F (x) + ::: + F (f n 1(x)) and the set of all limit points of the sequence ( n1 (F n(~x) x~))n2N is bounded. This set is the rotation set of x~ under F , denoted also by (F; x~). For a point x 2 T2, the set (F; x~) does not depend on the choice of x~ 2 1(x), so we can also write (F; x). In the case when (F; x) consists of one point we call it the rotation vector of x. This happens, for example, when x is periodic. More generally, if is an ergodic probability measure for f and x 2 T2 is typical for , then the Birkho ergodic theorem implies that R (F; x) = f F dg. The denition of (F; x) is in the spirit of Poincare's notion of the rotation number for a circle homeomorphism | for denitions and historical background one may consult [ALM93, Boy92, MZ89]. Much as in the case of degree-one circle maps ([NPT83]), the set (F; x~) generally changes as we vary x~. Unfortunately, the union Sx~2R (F; x~) often fails to be compact. For this reason, following [MZ89], we dene the rotation set of F as ! [ 1 n \ 2 (F ) := cl (F (~x) x~) : x~ 2 R R2: n n>m m>0 2 Without going into details, let us just note that other existing denitions of 3 the rotation set yield sets that are contained in the above and may at most lack some points in the boundary. (These can not be the extremal points either.) As the intermediate value theorem forces the rotation set for a degree-one circle map to be an interval ([NPT83]), the following result is ultimately a consequence of the Jordan simple curve theorem. Theorem 0.0.1 ([MZ89]) If F 2 H~ (T2) then (F ) is convex. Let us note that, while the denition of (F ) makes sense for non-invertible maps F , it renders then a set that is a continuum but often lacks convexity ([MZ89]). Theorem 0.0.1 makes one ask: Can any compact convex subset of R2 be realized as (F ) for some F 2 H~ (T2 )? This question is still open. Even deciding which line segments are rotation sets is not yet resolved. Given any interval with rational slope passing through a point with rational coordinates it is easy to nd a homeomorphism that has the interval as its rotation set. The construction in [FM90] implies that intervals with at least one rational endpoint and irrational slope can be realized as well. Based on their results for time-one maps of toral ows, Franks and Misiurewicz conjecture that these are the two only possibilities ([FM90]). In the case of nonempty interior, it is known that all polygons with vertices having rational coordinates can be realized as rotation sets ([Kwa92]). There is also an example with (countable) innity of the extremal points ([Kwa]). While the basic nature of the rotation set is not fully understood, there is a considerable number of results linking it to dynamical phenomena exhibited 4 by the map. The following theorem of Franks is a pillar of the progress in this direction. Theorem 0.0.2 ([Fra89]) If F 2 H~ (T2) and p 2 Z2 , q 2 N are such that p=q 2 int((F )), then F q (~x) = x~ + p for some x~ 2 R2. Moreover, if q and the two coordinates of p have no common divisor, then q is the least j 2 N with the property that F j (~x) x~ 2 Z2 . Remark. If v is an extremal point of (F ), then v = R F d for some ergodic measure ([MZ89]). Thus the assertion of the theorem is true also when p=q is extremal, by Theorem 3.5 in [Fra88]. In terms of the map f 2 H(T2 ) that is a projection of F , the above theorem says that any vector v in the interior of (F ) with rational coordinates is the rotation vector of a periodic point x = (~x) 2 T2 . Moreover, the period of this periodic point can be required to be equal to the smallest denominator q in the representation v = p=q, where p 2 Z2 , q 2 N. (Such periodic points are often called primitive.) Once again the assumption about invertibility of f is relevant; Barge and Walker have a non-invertible example with no periodic points and with the nonempty interior of the rotation set ([BW93]). One reason why Franks's result is important is that it facilitates application of very powerful methods of the Thurston-Nilsen theory ([Thu88, FLP79, Boy94]) to f 2 H(T2) with nonempty interior of the rotation set. Indeed, by puncturing T2 at the points of the orbits found by Theorem 0.0.2, one gets a surface of negative Euler characteristic with a homeomorphism that, not only does not have to be isotopic to the identity, but often is isotopic to 5 a pseudo-Anosov map (cf. Theorem 1.2.1). This trick, going back to Bowen ([Bow78a]), was applied by Llibre and MacKay and subsequently by Misiurewicz and Ziemian to prove, among other things, the following theorems. Theorem 0.0.3 ([LM91]) If f 2 H(T2), F 2 H~ (T2) is its lift, then, for any continuum C int((F )), there is a point x 2 T2 with (F; x) = C . Theorem 0.0.4 ([MZ91]) If f 2 H(T2) and F 2 H~ (T2) is its lift, then, for any v 2 int((F )), there is an invariant set K T2 such that (F; x) = v for all x 2 K . Thus nonempty interior of the rotation set implies uncountably many invariant sets each with a dierent non-collinear rotation vector. Analogous behavior for annulus homeomorphisms is exhibited by integrable twist maps; however, on the torus (as with degree-one circle maps), it is a sure sign of chaos. Theorem 0.0.5 ([LM91]) If f 2 H(T2), F 2 H~ (T2 ) is its lift and int(F ) 6= ;, then the topological entropy htop (f ) of f is positive. For degree-one circle maps there exist explicit and sharp lower bounds for the topological entropy in terms of the rotation set ([ALM93]); this prompted Llibre and MacKay to ask whether there is such a relation in the case of T2. We give an answer to this question | a precise formulation of the result can be found in the beginning of Chapter 1. Below, we give a simplied version that is formally still a conjecture but approximates our theorem very well. 6 \Theorem" 1 (cf. Theorem 1.1.1) There exist a universal constant C > 0 such that, if f 2 H(T2 ) and F 2 H~ (T2) is its lift, then (i) htop (f ) C ln Area((F )), for (F ) that are not \small"; q (ii) htop (f ) C Area((F )), for (F ) that are not \large". Our estimate is not sharp. (Nevertheless, it captures the right asymptotics for \large" rotation sets | see the example in Chapter 1.) We think of it more as an evidence of the power of our method than an isolated result. The method that we advocate takes over where Llibre and Mackay left o and exploits the connection between the topological entropy and the quasi-conformal dilatation of the underlying pseudo-Anosov map. This approach has a lot of potential and ultimately should lead to better results. The second main idea in this thesis is that of topological entropy at the rotation vector. Once again our motivation was Theorem 0.0.5 which made us ask: If int((F )) 6= ;, how much topological entropy is contributed by the points with a particular rotation vector? We make the rst step towards answering this question. Namely, for a map f 2 H(T2 ) and its lift F 2 H~ (T2), we dene for each v 2 (F ) the topological entropy at the rotation vector, denoted by (v) (v) htop (F ). We show that htop (F ) obeys a form of the thermodynamical formalism, which we then employ to give a complete analysis of the pseudo-Anosov case and, ultimately, strengthen \Theorem" 1 to the following: \Theorem" 2 (cf. Theorem 3.1.1) For some universal constant C > 0, if f 2 H(T2 ) is a dieomorphism, F 2 H~ (T2) is its lift and v 2 int((F )), then there exists a compact invariant set K T2 such that (F; x) = v, for all 7 x 2 K , and (v) htop (f ) htop (f jK ) C ln Area((F )) (v); for (F ) that are not \small", q (v) htop (f ) htop (f jK ) C Area((F )) (v); for (F ) that are not \large". The quantity (v) is a certain measure of the \distance" of v from the boundary of (F ). Ultimately, one should be able to make the choice of the set K above so (v) (f ). We prove this only for maps that htop (f jK ) is arbitrarily close to htop that are pseudo-Anosov relative to a nite set (see [Boy94] for the denition). Another important issue, which we resolve here only for that class of maps, (v) is the regularity of htop as a function of v. We have the following theorem (summarizing Proposition 3.2.2 and Theorem 3.2.1). Theorem 0.0.6 If f 2 H(T2) is pseudo-Anosov relative to a nite set and F 2 H~ (T2) is its lift, then (v) (i) htop (F ) is real-analytic and strictly concave on int((F )); (ii) for any v 2 int((F )), there exists a fully supported ergodic measure (v) R (v) such that the measure theoretic entropy h v (f ) = htop (F ) and F d(v) = v; ( ) (iii) for any v 2 int((F )) and 2 (0; 1), there exists a compact invariant (v) subset K of T2 such that htop (F jK ) htop (F ) and (F; x) = fvg for all x 2 K. 8 Actually, there is more in our analysis than the above theorem spells out; for (v) example, htop (F ) can be explicitly calculated from the Markov partition. The line of attack in proving the theorem is not original at all: we use symbolic dynamics to reduce it to facts about subshifts of nite type. Still, our discussion of subshifts of nite type contains some new results. In particular, the following theorem may be useful in many applications. Theorem 0.0.7 (cf. Theorem 2.2.4) Suppose : ! is a transitive subR shift of nite type and : ! Rd is locally constant. If v = d for some invariant ergodic probability measure , then, for any 2 (0; 1), there exists a compact invariant set K such that htop ( jK ) h ( ) and nX1 i (f (x)) nv < +1: sup n2N;x2K i=0 The above theorem for integer valued and integer v coincides with the re- sult in [MT92]. There, it is applied to calculate the minimal entropy of some sequences of patterns of periodic orbits for one-dimensional maps. The discussion in [MT92] yields concrete formulas only for symmetric patterns. Our methods (which are dierent) can be used to treat equally eciently the general case. In particular, the Gibbs states that we discuss realize the variational principle in [MT92] thus answering the question posed there. 1 (v) As a tool for comprehending htop we develop (in Chapter 2) a weak but quite general version of the thermodynamical formalism for a dynamical system on a compact space with a continuous observable in Rd. Theorem 2.1.1 1 The relation of my work to that of [MT92] was brought to my attention only recently. I will give the details elsewhere. 9 (v) plays the central role here by connecting the entropy htop with the appropriate pressure function. Even though this theorem can be viewed as a consequence of the standard thermodynamical formalism ([Wal82]), the implication is not trivial; to establish it, we generalize the variational principle for topological entropy ([Wal82]) by making it sensitive to the averages of the observable (see Theorem 2.1.2). This material should be thought of as just another realization of the paradigm of the thermodynamical formalism ([Rue78, Bow75, Wal82]). We hope that our presentation will help to absorb these ideas into the theory of rotation sets (including the context of other surfaces than torus, see [Fra94, Boy94].) (v) Let us also note that in dealing with htop one can get pretty far using elementary large deviation techniques. The inspiration for this approach comes from a book by Ellis ([Ell85]). In Appendix A we use it to give a proof of the thermodynamical formalism theorem (Theorem 2.1.1). Finally, our last chapter is devoted to a concrete family of torus homeomorphisms that are isotopic to the identity. The purpose of this discussion is two-fold. For one, we wanted to give a concrete physical model that satises the assumptions of our theorems. The other reason is mathematical: these are perhaps the simplest and most accessible analytically mappings exhibiting nonempty interior of the rotation set; even though studied from other points of view for quite some time, they somehow avoided to this date theoretical analysis with the emphasis on their rotation sets ([LL93, ZZRSC86b, CSUZ87, ZZRSC86a]). We give the beginning of such 10 an analysis. In particular, we prove monotonicity of the rotation set for a certain (physically) natural one-parameter family of torus maps. This may be interesting since such examples were not known before. Chapter 1 An estimate of entropy for toroidal chaos In this chapter we show that topological entropy can be estimated from below by looking at the \size" of the rotation set. This \size" can be roughly thought of as the area of the rotation set (but, in general, is not comparable with it). 1.1 Statement of the result Consider f 2 H(T2 ) and its lift F 2 H~ (T2). Assume that int((F )) 6= ;. It was proved in [LM91] that f exhibits then toroidal chaos; in particular, f has positive topological entropy htop (f ) > 0 (see Theorem 0.0.5). Our goal is to provide a lower bound on the entropy in terms of the size of the rotation set. To formulate the result, let us rst explain how we measure the size of the rotation set. 11 12 Denition 1.1.1 Let K R2 be convex and compact. If int(K ) contains three non-collinear points of Z2 , then we dene A(K ) := maxfpr r : r ; r 2 N and there exist p; v; w 2 Z2 such that 1 2 1 2 p; p + r1v; p + r2w 2 int(K ) and are non-collinearg; I (K ) := 1: Otherwise, we set A(K ) := 0; I (K ) := inf f 2 N : int(K ) contains three non-collinear points of Z2 g: Note that, for m 2 N, if A(K ) > 0, then A(mK ) mA(K ), and that A ( I ( K ) K ) 1: (1:1) Theorem 1.1.1 (Main Lower Bound ([Kwa93])) There exist universal constants C , C 0 such that, for any f 2 H(T2 ) and its lift F 2 H~ (T2 ), we have htop (f ) C ln+ A((F )); htop (f ) C 0=I ((F )): We think of the rst estimate as signicant for \large" (F ) and of the second as signicant for \small" (F ). Remark. In fact, we will prove that if r1 and r2 realizing A((F )) are both greater than 5 then q htop (f ) 31 ln (r1 1)(r2 1) 12 ln 4: 13 By considering the eighth iterate of F , this yields C = 1=27 and C 0 = ln 2=54 | see Claim 1.4.1 and the discussion opening Section 1.4. As indicated by the remark, our statement of the theorem is a compromise between sharpness of the result and complexity of its formulation. It will be clear from the proof that we are far from providing optimal estimates, so we see no point in complicating the inequalities. Nevertheless, the theorem is asymptotically sharp (up to a multiplicative constant) for \large" rotation sets, as shown by the following example. Example. For n 2 N, dene Un ; Vn 2 H~ (T2) by Vn(x; y) := (x+n sin(2y); y), Un(x; y) := (x; y + n sin(2x)), x; y 2 R. Consider Fn := Vn Un 2 H~ (T2) and the corresponding fn 2 H(T2). The points (0; 0); ( 41 ; 0); (0; 14 ) have rotation vectors (0; 0); (0; n); (n; 0) respectively, so A((Fn )) n. On the other hand, the maximal Lyapunov exponent dominates htop (fn) and is trivially bounded by ln sup krfnk ln(2n) . In this way, we have lim sup ln (htop (fn)=A((Fn))) 1: n!1 One would like to have an example of a family ffg>0, f 2 H(T2), with lifts fFg>0 having (F) shrinking to a point, as ! 0, and such that htop (f) I ((F)) Const, for > 0. This would mean that our estimate is asymptotically sharp for \small" rotation set. No such example is known to the author. The rest of this chapter is devoted to the proof of the theorem. We isolate the main steps into separate sections. Section 1.2 describes reduction to the 14 case of a pseudo-Anosov map. Section 1.3 explains how entropy is linked with the quasi-conformal dilatation of conformal structures on T2 . The very estimates proving the theorem can be found in Section 1.4. 1.2 Comparison with pseudo-Anosov maps We describe here a trick that allows one to pass from investigation of a general map in H(T2) with nonempty interior of the rotation set to a pseudoAnosov map with approximately the same rotation set. Our version is only a trivial extension of the original one in [LM91]. We use the notion of a pseudoAnosov map (of a surface) rel a nite invariant set. Referring the reader to [Boy94] for an exposition, we only say here that, if f : S ! S is a pseudoAnosov map rel a nite invariant set P , then it is a factor of a pseudo-Anosov map ([Thu88, FLP79]) on the surface obtained by compactifying the ideal boundary of S n P with circles. The factor map collapses the boundary circles to the corresponding points of P and is injective otherwise. Thus the dynamics of the two maps almost coincide. (In particular, they have the same topological entropy ([Bow78a]).) Theorem 1.2.1 Suppose that F 2 H~ (T2) and int((F )) 6= ;. Let f 2 H(T2) be the map that lifts to F . If 0 int((F )) is a convex non-degenerate polygon with vertices in Q2 , then there exists a nite f -invariant set P T2 and a pseudo-Anosov rel P map g 2 H(T2 ) that is isotopic to f rel P and has a lift G 2 H~ (T2) with 0 (G). 15 Remark. The above theorem is also true when 0 has some vertices on the boundary of (F ) as long as these are extremal points of (F ) (cf. the remark after Theorem 0.0.2). Pseudo-Anosov maps have minimal topological entropy in their isotopy class ([Thu88, FLP79]), so htop (f ) htop (g) > 0: (1:2) Remark. Pseudo-Anosov maps not only minimize entropy but also have \the simplest dynamics" in their isotopy class ([Han85, Boy10]). In particular, we have (G) (F ). (For details and proofs see Theorem 3.3.1 and the corollary after it.) Proof of Theorem 1.2.1. Let v1 ; :::; vm 2 Q2 be the vertices of 0 (or, equally well, any nite set in Q2 such that 0 = conv fv1; :::; vm g, where conv abbreviates convex hull). Write vi = pi=qi, where pi 2 Z2 , qi 2 N and the two components of pi and qi have no common divisor, all this for i = 1; :::; m. By Theorem 0.0.2, there exist x1 ; :::; xm 2 R2 such that F qi (xi ) = xi + pi, i = 1; :::; m. Let P be the union of the projections on the torus of the orbits of x1 ; :::; xm . If g 2 H(T2) is isotopic to f rel P and G 2 H~ (T2) is the lift of g equivariantly isotopic to F relative 1(P ), then 0 = convfv1 ; :::; vmg (G). In particular, we can take for g the Thurston canonical representative of the isotopy class of f relative P (see e.g. Th. 7.1 in [Boy94]). Following [LM91], one can argue that g can not have any reducing curves, and so it is pseudoAnosov. Indeed, since a reducing curve must be non-peripheral, we have a priori three possibilities: it is essential on T2 , and then (G) would have to 16 be contained in a line; it bounds a disk containing points of dierent orbits in P , which would contradict the fact that the rotation vectors of these orbits are dierent; it bounds a disk containing two or more points of one orbit in P , which would force pi and qi to have a common divisor for some i, contrary to the assumption. Q.E.D. 1.3 Entropy and conformal structures It is clear from the previous section that, to prove the Main Lower Bound, we merely have to correlate the size of the rotation set with the topological entropy for pseudo-Anosov maps. A natural rst impulse is to try linking the two via standard symbolic dynamics given by the Markov partitions. This however leaves one with a task of understanding how topology of the torus inuences the structure of the corresponding subshifts of nite type. We propose here a dierent approach. Namely, we will take advantage of the fact that the the entropy of the pseudo-Anosov map g can be calculated by looking at its quasi-conformal dilatation with respect to a preferred family of conformal structures on T2. We explain this below. (Following [Alh66], we abbreviate quasi-conformal to q.c.) Denote by C (T2 n P ) the set of all conformal structures on T2 n P . If 2 C (T2 n P ) and : T2 n P ! T2 n P is a -q.c. map, then we write K [ ] for the (maximal) q.c. dilatation of ([Alh66]). (For conformal maps K = 1.) According to Bers (Ch. III, [Abi80]), there exists g 2 C (T2 n P ) solving the 17 following extremal problem Kg [g] = inf fK [ ] : 2 C (T2 n P ); isotopic to gg: (1:3) Moreover, g is modeled around each boundary component of T2 n P on a punctured disc (see Th. 3, x2, Ch. III, [Abi80]). Thus g extends uniquely through the punctures to a conformal structure on T2 which, with no risk of confusion, is also denoted by g . For the sake of giving a more intuitive picture, let us sketch an alternative (and more geometrical) way of nding g . From the denition of pseudoAnosov map relative a nite set (see [Boy94]), there exists a pair of transverse measured singular foliations F s and F u such that g(F s) = 1F s and g(F u) = F u, where = exp htop (g). The two foliations dene a smooth structure together with a at Riemannian metric on the complement of the set of singularities Q T2 (P Q). This Riemannian metric naturally determines a conformal structure on T2 n Q. However, due to the form of F s and F u at the singularities that is postulated by the denition, this structure has removable singularities at points of Q and so determines a unique conformal structure g on T2. From the action of g on F s and F u one sees the q.c. dilatation to be ([Abi80]) K [g] = 2 = exp(2htop (g)): Putting this together with inequality (1.2) we get h(f ) (1=2) ln Kg [g]: (1:4) 18 In our applications we will not use any a priori knowledge of g , thus, weakening (1.4), we state the following theorem. Theorem 1.3.1 If f 2 H(T2) and P is a collection of primitive periodic orbits with dierent non-collinear rotation vectors, then htop (f ) inf f(1=2) ln K [ ] : 2 C (T2 ); -q.c. map of T2 isotopic to f rel P g: Let us note here a curious problem related to the above theorem. Open Problem. If S is a compact Riemann surface and f : S ! S is a q.c. homeomorphism with q.c. dilatation K [f ], is htop (f ) (1=2) ln K [f ] ? For a C 1-smooth map, the answer is yes. Indeed, by the variational principle for topological entropy ([Man87]), we just need to prove the following: if is an ergodic invariant measure, then h (f ) ln K [f ]=2. Consider the Lyapunov exponents + of . By Ruelle's inequality ([Man87]), we can write h (f ) maxf0; +g and h (f ) = h(f 1) maxf0; g. Thus either h (f ) = 0, and there is nothing to prove, or h (f ) minf+; g (+ )=2. By looking at the derivative Df n(x) at a typical point x and for large n, we get exp += exp K [f ], which already implies our claim. In the non-smooth case, there are estimates by Gromov with a non-optimal multiplicative constant ([Gro87]). 19 1.4 Proof of the Main Lower Bound We have two inequalities to prove htop (f ) C ln+ A((F )); (1:5) htop (f ) C 0=I ((F )): (1:6) and In fact, (1.6) follows from (1.5). For a proof, set := I ((F )) 2 N. Using (1.1), we get A((F 2 )) = A(2(F )) 2. Application of (1.5) to f 2 yields 2 htop (f ) = htop (f 2 ) C ln 2. The inequality (1.6) with C 0 = (C ln 2)=2 follows. In the rest of this section we tackle (1.5). Let p; v; w 2 Z2 and r1 ; r2 2 N be as in the denition of A((F )). Passing perhaps to the sixth iterate of F , we may assume that r1 ; r2 6. Also, it is easy to see that we can require that v; w form a basis of Z2 as a Z-module. By the theorem of Franks (Theorem 0.0.2), for each of p; p + r1 v; p + r2w 2 (F ) \ Z2 , there is a xed point of f realizing it as the rotation vector. Let P be the union of those three xed points. By Theorem 1.3.1, to estimate htop (f ), we need to consider an arbitrary conformal structure 2 C (T2 ) and a -q.c. map g : T2 ! T2 which is isotopic to f rel P . By the uniformization theorem ([Abi80]), the conformal torus (T2; ) is biholomorphic to C= , where is a rank-two lattice in C. The map g lifts to a map : C ! C which is q.c. with respect to the standard conformal structure on C with the q.c. dilatation K [] = Kg [g]. Let e1 ; e2 2 be the vectors corresponding to the vectors v; w. After perhaps post-composing 20 with a deck transformation, the three xed points in P lift to p0 ; p1; p2 2 C such that (p0 ) = p0; (p1 ) = p1 + r1 e1; (p2) = p2 + r2 e2: (1:7) Our task is to verify, from the above topological data on , the following claim which already implies (1.5). Claim 1.4.1 (target estimate) If r1; r2 6, then K [] 41 (r1 1)1=3(r2 1)1=3 : (1:8) The proof of the claim is rather technical. To better separate the ideas, after necessary denitions, we formulate the key observations as separate Facts and then prove them. All the pieces of the puzzle are put together in the nal argument at the end of this section. For any ordered pair (a; b) of orthogonal vectors in C, we have the rectangle R = R(a; b) = fxa + yb : 0 x; y 1g. For any s 2 [0; 1], let s[R] : [0; 1] ! R be given by s[R](t) = ta + sb, and s[R] : [0; 1] ! R be given by s[R](t) = sa + tb. Thus we have two families of curves lling R: a-family A := fy [R]gy2[0;1], and b-family B := fx[R]gx2[0;1]. Clearly, @R = 0[R] [ 1[R] [ 0[R] [ 1[R]. We call the rectangle R fundamental for if every orbit under of intersects its interior exactly once. All the above denitions extend in an obvious way to any translate of the rectangle R. For a family G of rectiable curves in C, we denote the extremal length of G by (G ), i.e. ZZ (G ) := sup L()2 : : C ! R+ measurable with 2dxdy = 1 ; 21 where L() := inf Z jdzj : 2 G : For the Euclidean length of curve we simply write l( ). For the families A and B we have ([Alh66]) (A) = jaj2=jRj; (B) = jbj2 =jRj; (1:9) where jRj stands for the Euclidean area of R. Our estimation of K [] will be based on the quasi-invariance of the extremal length, a property encapsulated by the following inequality ([Alh66]) K [] (G )=(G ): (1:10) For G above we will take a-families and b-families of appropriate rectangles. The following simple lemma plays a key role in nding \good rectangles". Lemma 1.4.1 If v; w 2 C are orthogonal, v 2 and R = R(v; w) is fundamental for , then there exists z 2 C, for which R0 := R + z satises (A) l(0)2 =jR0j = l(1)2=jR0j; (1:11) (B) l(0 )2=jR0j = l(1 )2=jR0j; (1:12) where A = fy g, B = fxg are the a-family and b-family for R0 . Proof. First slide R along the line Rw to nd t0 such that l((0 [R] + t0 w)) = inf fl((0 [R] + tw)) : t 2 Rg: 22 This means that l(0[R + z]) = inf fl() : 2 a-family of R + zg; (1:13) where z = t0 w. Observe that (1.13) holds also for any z = t0w + sv, s 2 R, because for any s1 ; s2 2 R the curves y [R + t0 + siv], i = 1; 2, project to the same loop on the torus and so their images under have equal length. In this way, if sliding along Rv we have found s0 such that l(0 [R + z]) = inf fl( ) : 2 b-family of R + zg; z = t0 w + s0 v; (1:14) we got both (1.13) and (1.14) satised for z = t0w + s0v. Set R0 := R + z. The inequalities in (1.11) and (1.12) follow from the denition of the extremal length when one takes for a multiple of the characteristic function of R0 while remembering that jR0j = jR0j. The equalities are trivial since 0 and 1, as well as 0 and 1 , project to the same arcs on the torus. Q.E.D. Let 2 [0; ) be the measure of the convex angle between e1 and e2 . Fact 1.4.1 For i; j 2 f1; 2g, i 6= j , we have 8(jeij2 + sin2 jej j2)K [] (ri 1)2jeij2 + (rj 1)2 sin2 jej j2 + maxf0; rj jej jj cos j jeijg2: Proof. Let v := ei and w be the component of ej orthogonal to ei. Let R0 be as in Lemma 1.4.1. Due to (1.7), (R0) intersects R0 , R0 + riei, R0 + rj ej , so l((@R0 )) dist(R0; R0 + riei ) + dist(R0 ; R0 + rj ej ); 23 and consequently, (using the Pythagorean theorem), l((@R0 )) (ri 1)jeij (1.15) 1=2 + (rj 1)2jej j2 sin2 + maxf0; rj jej jj cos j jeijg2 : Feeding (1.9), (1.11), (1.12) into (1.10), we obtain (jeij2 + sin2 jej j2)K [] jeij2 (A)=(A) + jej j2 sin2 (B)=(B) 2 2 jeij2 l(jRj0 ) l(jRj)2 + jej j2 sin2 l(jRj0 ) l(jRj)2 0 0 2 2 = l(0 ) + l(0 ) 21 (l(0 ) + l(0))2 = 81 l((@R0 ))2 : This inequality put together with (1.15) proves Fact 1.4.1. Q.E.D. Note that, if the lengths of e1 and e2 were \comparable", Fact 1.4.1 would yield an estimate similar to (1.8) of Claim 1.4.1. The following fact exhibits one mechanism preventing the je1 j and je2 j from being too disproportional. Fact 1.4.2 If d 2 (0; 1) and e1 = 1, e2 = dp 1, = Z + dp 1 Z, then d2K [] (r1 1)2: Proof. Let R0 be the rectangle obtained by the application of Lemma 1.4.1 p p with v := 1 and w := d 1. Set := Sf1 + kd 1 : k 2 Zg. Clearly, is a straight line cutting C in two half-planes. Denote by H the halfplane containing R0 and by H + the one containing R0 + 1. The rectangle R0 24 is fundamental, so, by (1.7), there are q0; q1 2 R0 such that (q0 ) = q0 and (q1) = q1 + r1 : Observe that, because q0 + 1 = (q0 + 1) 2 (H +), we have q0 = (q0 ) 2 (H ) and q0 + 1 = (q0 + 1) 62 (H ); and, because q1 + r1 + 1 = (q1 + 1) 2 (H +), also q1 + r1 = (q1) 2 (H ) and q1 + r1 + 1 62 (H ): Thus straight line segments in C, [q0 ; q0 + 1] and [q1 + r1 ; q1 + r1 + 1], both intersect ( ) = @ ((H )). Consequently, if prx is the orthogonal projection on the real axis R C, then diam(prx( )) r1 1. This, using -equivariance of , yields l(1 ) diam(prx(1)) = diam(prx( )) r1 1: Now we apply (1.9), (1.10) and (1.12) to get K [] (B)=(B) l(1)2=d2 (r1 1)2=d2; and we are done. Q.E.D. Fact 1.4.3 Suppose that e1 = 1, e2 = l cos + l sin p 1, l 2 (0; 1). If sin2 (r1 1)2=(32K []); then K [] 41 (r1 1)1=3(r2 1)1=3 : 25 Proof. Put d := l sin . First assume that = =2. Fact 1.4.1, applied with p 1), gives us i = 1, j = 2 (i.e. ei = 1 and ej = d 8(1 + d2)K [] (r1 1)2 + (r2 1)2d2 (r2 1)2d2: Inserting in the above inequality the estimate of d given by Fact 1.4.2, we derive 8 2 K [] (r2 1)2(r1 1)2=K []; and nally, K [] 14 (r1 1)(r2 1): (1:16) In general, 6= =2 and then we consider := L L 1 , where p p L : C ! C is given by L (x + y 1) := x + cot y + y 1 and transforms p p the lattice Z + d 1Z to Z + (l cos + l sin 1)Z. The map L is areapreserving, so K [L ] equals the largest eigenvalue of L L , where L is the adjoint of L . The trace of L L is 2 + cot2 , so we have K [L ] 2 + cot2 : Observe that (1.16) holds once is replaced by . Since by sub-multiplicativity of q.c. dilatation K [] K []=K [L ]2; we get K [] 14 (r1 1)(r2 1)=(2 + cot2 )2 : (1:17) Our hypothesis on can be written as cot2 sin 2 32K []=(r1 1)2: (1:18) 26 If cot2 1, then, by (1.17) and using r1; r2 6, we see that 2=3 2=3 K [] 4 132 (r1 1)(r2 1) 5 4 59 (r1 1)1=3(r2 1)1=3 14 (r1 1)1=3 (r2 1)1=3 : If cot2 1, then, by (1.17) and (1.18), we obtain K [] 14 (r1 1)(r2 1)=(3 cot2 )2 4 321 322 (r1 1)5(r2 1)=K []2; which yields K [] ! (r1 1)4 1=3 (r 1)1=3 (r 1)1=3 1 2 4 32 322 !1=3 54 1=3 1=3 1=3 1=3 1 ( r 1 1) (r2 1) (r1 1) (r2 1) : 2 2 4 3 32 4 Q.E.D. Conclusion of the proof of Claim 1.4.1. We may assume that je1 j je2 j, because otherwise one can switch the two. With the aid of a linear conformal p map we can further adjust e1 , e2 to e1 := 1 and e2 := l cos + l sin 1, where l 2 (0; 1]. For \large" , namely when sin2 (r1 1)2=(32K []); we are done by Fact 1.4.3. Assume then that is \small", that is sin2 (r1 1)2=(32K []): Apply rst Fact 1.4.1 with i = 1 and j = 2 to see that 16K [] 8(1 + sin2 l2 )K [] (r1 1)2; 27 and so sin2 1=2. Thus, using r1 6, we verify that q q r1 je1jj cos j je2j 6 1 sin2 l 6 1=2 1 > 0; and application of Fact 1.4.1, with i = 2, j = 1, yields 8(l2 + sin2 )K [] (r2 1)2l2 + (r1 1)2 sin2 + (r1j cos j l)2 (r2 1)2l2 + (r1 1)2 2r1j cos j (r2 1)2l2 + (r1 1)2=2; (1.19) where for the last inequality we used r1 6 again. Thus, dropping (r2 1)2l2 and isolating l2 , we get l2 (r1 1)2=(2 8K []) sin2 (r1 1)2=(32K []) sin2 ; and, applying (1.19) again, we derive K [ ] (r2 1)2l2 =(16l2) + (r1 1)2=(32l2) (r2 1)2=16 + (r1 1)2=32 1 (r 1)(r 1) 2 16 1 1 (r 1)1=3(r 1)1=3: 2 4 1 Q.E.D. Chapter 2 Topological entropy at the asymptotic average Our ultimate goal in this work is to discuss topological entropy relative to a vector in the rotation set. A large part of our discussion naturally ts in a more general context, so we stray from the main course in this chapter. In the rst section we dene topological entropy at the asymptotic average for a map of a compact metrizable space with a continuous observable. We prove the corresponding variational principle and use it for a rather weak version of the thermodynamical formalism, which holds in this generality. Section 2.2 deals with subshifts of nite type. Invoking the standard transfer matrix trick, we give a real-analytic formula for the topological entropy at the asymptotic average. More importantly, we develop a technique that enables us to construct, for any given asymptotic average, compact invariant sets \almost realizing" the topological entropy at this asymptotic average. 28 29 2.1 Relative variational principle and thermodynamical formalism Consider a compact metrizable topological space X with metric d and a dynamical system generated by a homeomorphism f : X ! X . Moreover, assume that a continuous function (called an observable ) : X ! Rd is given. Our primary example will be a torus homeomorphism isotopic to the identity with the displacement function of one of its lifts as the observable. The rotation set becomes then a specialization of the set of asymptotic averages (f; ) dened by (f; ) := \ m>0 cl [ n>m ! f(1=n)Sn(f; )(x) : x 2 X g Rd; (2:1) where Sn(f; ) := (x) + ::: + (f n 1(x)) is the Birkho sum (cf. [Zie95]). The set (f; ) is compact. Abstracting the proof of connectedness of the rotation set in [MZ89], one can see that (f; ) is connected for connected X . The complexity of the dynamics detected by an asymptotic average in (f; ) may be measured by the associated topological entropy. In dening this entropy one may follow a few paths leading to dierent invariants which nevertheless coincide for the important examples of our primary concern (e.g. for pseudo-Anosov homeomorphisms). Let us start with the most elementary approach building directly on Bowen's denition of topological entropy (see [Wal82]). Given S X , we say that S is (; n)-separated if, for every two dierent x; y 2 S , we have d(f k (x); f k (y)) , for some k 2 f0; 1; :::; n 1g. For a set 30 E Rd and > 0, we dene sE (f; ; ; n) := maxf#S : S X (; n)-separated, 8x2S (1=n)Sn(f; )(x) 2 E g; and, under the convention that ln 0 = 1, we set hE (f; ; ) := lim sup n1 ln sE (f; ; ; n) 2 f 1g [ [0; 1]: n!1 The topological entropy associated with E is then hEtop (f; ) := lim hE (f; ; ): !0 Notice that hE (f; ; ), as a function of , is decreasing, so the limit actually exists. Also, as suggested by the notation, hEtop (f; ) is a topological invariant; it does not depend on the metric, nor is altered by replacing the pair (f; ) with a (h 1 f h; h), where h is a homeomorphism of X (cf. Proposition 2.1.2). Actually, one can easily make hEtop manifestly topologically invariant by rephrasing its denition in terms of open covers, as it is done for the usual topological entropy (see [Wal82]). Also, hEtop (f; ; ) is monotonic in E and, for two sets E; F 2 Rd, we have n o E [F (f; ) = max hE (f; ); hF (f; ) : htop top top (2:2) Thus a bulk of information carried by hEtop (f; ; ) is still recoverable from topological entropy at the asymptotic average, which is dened as the following function of v 2 Rd (v) htop (f; ) := rlim hBr (v) (f; ); !0 top where Br (v) is the open ball of radius r centered at v. 31 Actually, if we regularize E 7! hE (f; ) by setting (E ) htop (f; ) := rlim hBr (E)(f; ); !0 top (v) (E ) (v) then v 7! htop (f; ) is characterized by the relation htop (f; ) = maxfhtop (f; ) : v 2 E g, where E is a closed subset of Rd (cf. Lemma A.0.1). (v) The following is a summary of basic properties of htop (f; ) that can be easily derived from the denition | see also the appendix. In fact, part (ii) (v) below expresses the compatibility of htop (f; ) and (f; ) that was one of the (v) axioms lying at the origin of htop (f; ). Proposition 2.1.1 (basics) With the above denitions, we have (v) (i) 1 htop (f; ) htop (f ), v 2 Rd; (v) (ii) (f; ) = fv 2 Rd : htop (f; ) 0g; (v ) (iii) there exists v 2 Rd such that htop (f; ) = htop (f ); (v) (iv) htop (f; ) is upper semi-continuous in v, i.e. (v) (v) lim sup htop (f; ) = htop (f; ): x!v Proof of Proposition 2.1.1. Parts (i), (ii), (iv) require no comment. To see (iii), note that (2.2) implies that, for any nite cover of (f; ), there exists an element E of the cover with hEtop (f; ) = htop (f ). To nail down v simply consider ner and ner covers (cf. Lemma A.0.1). Q.E.D. (v) There are many properties of htop (f; ) that are directly analogous to those of the topological entropy, and the proofs are typically just simple modications of the standard ones. We mention here only those properties that we will need in the further development. 32 Proposition 2.1.2 (Quotient Rule) Let (X; d) and (Y; e) be compact metric spaces and : X ! Rd . If f : X ! X is a factor of g : Y ! Y via h : Y ! X (i.e. h is continuous surjective and f h = h g) then, for any (v) (v) E Rd, hEtop (f; ) hEtop (g; h). Consequently, htop (f; ) htop (g; h), v 2 Rd . Proof. Fix an arbitrary > 0. By uniform continuity of h, there is > 0 such that e(y; y0) < implies d(h(y); h(y0)) < for all y; y0 2 Y . Now, given S X which is (; n)-separated with Sn(f; )(x) 2 E for x 2 S , form S 0 Y by choosing a point from h 1 (x) for each x 2 S . This set is (; n)-separated for g and, for y 2 S 0, Sn(g; h)(y) = Sn(f; )(h(y)) 2 E . Since #S 0 = #S , we get hE (f; ; ) hE (g; h; ) hEtop (g; h), which nishes the proof because of arbitrariness of . Q.E.D. Proposition 2.1.3 (Power Rule) For any m 2 N and E Rd, we have hEtop (f m; (1=m)Sm()) = m hEtop (f; ): In particular, for v 2 Rd , (v) m (v) htop (f ; (1=m)Sm()) = m htop (f; ): Proof. This is a straightforward upgrade of the corresponding argument in [Wal82]. We trace it for the sake of completeness. Fix E Rd. We will prove that m hEtop (f; ) = hEtop (f m; (1=m)Sm(f; )). If S is an (; n)-separated set for f m, then it is (; nm)-separated for f . Moreover, (1=n)Sn(f m; (1=m)Sm(f; ))(x) = (1=(mn))Smn (f; )(x); (2:3) 33 for any x 2 X . It follows that hEtop (f m ; (1=m)Sm()) m hEtop (f; ). On the other hand, for > 0 there is > 0 such that d(x; y) forces d(f i(x); f i(y)) < , for 0 i m. Let S be a (; n)-separated set for f m maximal among those that satisfy (1=n)Sn(f m; (1=m)Sm(f; ))(x) 2 E for all x 2 S . By the maximality and (2.3), for any y 2 X with (1=(mn))Smn (f; )(y) 2 E , there is x 2 S so that d(f km(x); f km(y)) < , 0 k n 1, and so d(f i(x); f i(y)) < , 0 i mn 1. Consequently, if S 0 is an arbitrary (2; mn)-separated set such that (1=(mn))Smn(f; )(x) 2 E , x 2 S 0, then it has at most #S elements. By arbitrariness of S 0, one gets hE (f m; (1=m)Sm(); ) m hE (f; ; 2). It follows that hEtop (f m ; (1=m)Sm()) m hEtop (f; ). Q.E.D. The next theorem, in the spirit of thermodynamical formalism ([Rue78]), (v) is a powerful tool for calculating htop (f; ). The central role is played by the restriction of the topological pressure functional P ([Wal82]) to the linear space of functions fht; ()igt2Rd , where h; i is a scalar product in Rd. Namely, for every t 2 Rd, we consider t) (f; ) := p(top lim lim sup (1=n) ln max fPx2S exphSn(f; )(x); ti : S (; n)-separatedg : !0 n!1 (t) Note that p(0) top (f; ) = htop (f ) and that htop (f ) = +1 implies ptop (f; ) = +1 t) (f; ) < +1 for all t 2 Rd ; for all t 2 Rd. However, if htop (f ) < +1, then p(top this is the situation we are mainly interested in. t) (f; ) is a convex function of t. ConSince P is a convex functional, p(top vexity will play a signicant role in our considerations, so let us x some 34 denitions before we go further. Our main reference is [Roc72]. For a convex subset D Rd, denote by a(D) its ane hull, i.e. the smallest ane subspace of Rd containing it. The interior of D with respect to the natural topology on a(D) will be called relative interior and denoted ri(D). If g : Rd ! ( 1; +1] is an arbitrary function, epi(g) := f(x; y) : y g(x)g is the epigraph of g, and the set dom(g) := fx 2 Rd : g(x) < +1g is called the eective domain of g. The closure of g is the function cl(g) dened by cl(g)(y) := lim infx!y g(x); it is the greatest lower semi-continuous function majorized by g, and the epigraph of cl(g) is the closure of the epigraph of g. The convex conjugate g^ : Rd ! [ 1; +1] is dened by g^(y) := supfhx; yi g(x) : x 2 Rdg. The convex conjugate is a lower semi-continuous convex function. If g is a convex function, we have g^(y) = supfhx; yi g(x) : x 2 ri(dom(g))g (see [Roc72] p.104). Convex conjugacy has an involutive property; namely, (^g)^ = cl(convex hull(g)), where convex hull (g) is the pointwise supremum of all convex functions bounding g from below. The epigraph of convex hull(g) does not have to be equal to the convex hull of that of g. However, this is the case when g is lower semi-continuous and has bounded eective domain (see Proposition A.0.3 and its proof). Then also h := convex hull(g) is lower semicontinuous and h coincides with g at the extremal points of the epigraph of h. In particular, g(x) = h(x) at the ordinates x of those extremal points that lie over ri(dom(h)) | those values of x are referred to as points of strict convexity of h. Finally, if we assume that htop (f ) < 1, then all the convex functions in the following discussion are proper, i.e. they take values in ( 1; 1]. 35 Theorem 2.1.1 (thermodynamical formalism) If htop (f ) < 1, then the (v) (v) convex hull of v 7! htop (f; ) is the convex conjugate v 7! p^top (f; ) of t 7! t) (f; ). In particular, p(top t) (f; ) ; (v) htop (f; ) supd ht; vi p(top t2R and the equality holds at all points v where the right side is strictly convex. Using (ii) of Proposition 2.1.1, we get the following straightforward corollary. Corollary 2.1.1 If htop (f ) < 1, then the convex hull of the set of asymptotic (v) averages (f; ) is equal to the essential domain dom(^ptop (f; )) of the convex t) (f; ). conjugate of p(top Theorem 2.1.1 may be viewed as a large deviations type result (see [Ell85]). It was inspired by Theorem II 6.1 in [Ell85]. In the appendix we give a broader exposition together with an elementary proof. In this chapter we show Theo(v) rem 2.1.1 as a byproduct of connecting htop (f; ) with the main body of ergodic theory. The major link in the connection is the following improvement of the variational principle for topological entropy ([Wal82]). Denote by M(X; f ) the set of all Borel probability measures invariant under f and by E (X; f ) the set of all those measures in M(X; f ) that are ergodic. Theorem 2.1.2 (Relative Variational Principle) We have: (i) for any closed convex set V Rd , Z hVtop (f; ) sup h (f ) : 2 M(X; f ); d 2 V ; 36 (ii) for any open set U Rd , hUtop (f; ) sup Z h (f ) : 2 E (X; f ); d 2 U : Thus, if we dene (using the convention that sup ; = 1) and Z hverg (f; ) := sup h(f ) : 2 E (X; f ); d = v ; (2:4) Z hvmeas(f; ) := sup h(f ) : 2 M(X; f ); d = v ; (2:5) then we have the following corollary. Corollary 2.1.2 For v 2 Rd, we have (v) (f; ) lim sup hxmeas(f; ): lim sup hxerg (f; ) htop x!v x!v In particular, cl ( hvmeas(f; )) := lim sup hxmeas(f; ) = convex hull x!v (v) htop (f; ) : Proof of Corollary 2.1.2. The inequalities follow immediately from the deni- tions. To justify the second assertion, note that, since the metric entropy h(f ) is ane as the function of , hvmeas (f; ) is a convex function of v. From the ergodic decomposition theorem, hvmeas(f; ) is the convex hull of hverg (f; ). (v) We are done because htop (f; ) is pinched between the two. Q.E.D. Also, by its convexity, hvmeas(f; ) is continuous at all points v in the relative interior of its eective domain, ri((f; )); so, for those points, the rightmost \lim sup" in Corollary 2.1.2 is superuous. Another instance when 37 we do not need that \lim sup" is when h (f ) is upper semi-continuous in because then hvmeas(f; ) is upper semi-continuous in v. This happens for expansive f ([Wal82]) and also for any f that is C 1-smooth ([New89]). The following example shows that one can indeed have lim supx!v hxerg (f; ) (v) < htop (f; ). Example. For (xi )i2Z 2 f1; 2; 3; 4gZ, (xk ; :::; xl ) is a parity block if xk ; :::; xl are all even or all odd. We call it maximal if xk 1; xl+1 have the parity opposite to that of xi, i = k; :::; l. Consider a subshift (; ) where = fx 2 f1; 2; 3; 4gZ : x has at most two maximal parity blocksg and shifts to the left: (((xi)i2Z ))i := xi+1. The space is compact and invariant under . Let the observable : ! R be given by (x) := ( 1)x . One can easily verify that (; ) = [ 1; 1]. All ergodic invariant measures are supported on the union of even := fx 2 : xi even for all ig and odd := fx 2 : xi odd for all ig because any recurrent point is contained in one of the two sets. Nevertheless, h(0) top (; ) ln 2. Indeed, for any n 2 2N, consider points of with maximal parity blocks of the form (:::; xn=2 1 ) and (xn=2 ; :::). They all have vanishing Sn(; ), and there are 2 2n such points with pairwise dierent blocks of the rst n symbols. Perhaps a more interesting example is := fx 2 f1; 2; 3; 4gZ : 0 x has no two nite maximum parity blocks of equal lengthg: We leave analysis of this example to the reader. 38 Corollary 2.1.3 If 0 is an ergodic equilibrium state for ht0 ; i, that is ([Rue78, Bow75, Wal82]) D (t0 ) ptop (f; ) = h0 (f ) + t0 ; R Z E d0 ; v ) (f; ) = hv (f; ) = hv (f; ) = h (f ). then, for v0 := d0, we have h(top maes erg 0 0 0 0 In particular, if is an ergodic measure of maximal entropy, then v := R d satises h(v ) (f; ) = h (f ) realizing thus the maximum of h(v )(f; ). top top top Such a measure exists for C 1 maps due to the result by Newhouse ([New89]). Proof of Theorem 2.1.1 from Theorem 2.1.2. From the variational principle for topological pressure ([Wal82]) we have t) (f; ) = sup p(top Z h (f ) + h; ti d : 2 M(X; f ) = sup fht; vi + hvmeas(f; ) : v 2 dom ( hvmeas(f; ))g : t) (f; ) = ( hv (f; ))^, so, by the involutive property of \^" and by Thus p(top meas Corollary 2.1.2, we have t) (f; ))^ = ( hv (f; ))^^ = cl( hv (f; )) = convex hull h(v) (f; ) : (p(top top meas meas (v) (We skipped \cl" before convex hull htop (f; ) because this function is already lower semi-continuous | see Proposition A.0.3.) Q.E.D. Proof of Corollary 2.1.3. In the previous proof we got (v) p^top (f; ) = cl( hvmeas(f; )); v 2 Rd : (2:6) 39 Thus, combining trivial inequalities, we can write cl ( hvmaes(f; )) hvmaes(f; ) hverg (f; ) h (f ) = 0 0 0 n D Z E t) t) (f; )o = p^(v) (f; ) = t0 ; d0 p(top (f; ) supd ht; vi p(top top 0 0 t2R cl ( hvmaes(f; )) : 0 (v) In this way, cl ( hvmaes(f; )) = hverg (f; ). By Corollary 2.1.2, htop (f; ) is pinched between the two, which nishes the proof. Q.E.D. 0 0 Remark. If p(topt) (f; ) is dierentiable at t0 , then Corollary 2.1.3 is true without the hypothesis that 0 is ergodic. In fact, almost all measures in the R R ergodic decomposition of 0 satisfy then h (f ) = h (f ) and d = d0. R Indeed, ane functions : t 7! h (f ) + ht; d i are all majorized by t) (f; ) and, integrated with respect to , yield which is tangent to p(top t) (f; ) at t . Since there is a unique tangent at t , almost all 's must be p(top 0 0 0 0 equal to which implies our claim. 0 The following fact will be used numerous times ahead. Proposition 2.1.4 (Quotient Rule) Let (X; d) and (Y; e) be compact metric spaces and : X ! Rd be continuous. If f : X ! X is a factor of g : Y ! Y via h : Y ! X (i.e. h is continuous surjective and f h = h g) then, hverg (f; ) hverg (g; h), v 2 Rd . Proof. Let M(Y ), M(X ) be the spaces of the Borel probability measures on Y and X respectively, both equipped with the weak -topology. The map h induces the push forward operator on measures, h : M(Y ) ! M(X ). Since, 40 for 2 M(Y; g), hh() (f ) h (G), it is enough to prove that h maps E (Y; g) surjectively onto E (X; f ). This can be derived (via ergodic decomposition) from the lemma 8.3 in [Man87]. We give a more elementary argument due to Milnor. Fix an arbitrary 2 E (X; f ). Choose measures n 2 M(X ) supported on nite sets and converging to . By surjectivity of h, each atom of n has a preimage in Y ; and, by taking those preimages with the obvious weights, one easily gets n 2 M(Y ) such that h(n) = n. Passing perhaps to a subsequence, we can assume that n's converge to a measure 2 M(Y ). Since h is continuous, h ( ) = . To make invariant under g, replace with any limit point of the sequence of time averages (1=n) Pin=11 gi ( ) n2N | those averaged measures also map under h to . Thus we got 2 M(Y; g) such that h ( ) = . This shows that a compact convex set C := h 1 () \ M(Y; g) is not empty; therefore, it has an extremal point 2 M(Y; g). The measure must be ergodic. Indeed, if it were a nontrivial convex combination of two other measures 1; 2 2 M(Y; g), then, by extremality of in C , h(i ) 6= , i = 1; 2. This contradicts ergodicity of because = h ( ) is a convex combination of h(1 ) and h(2 ). Q.E.D. Proof of Theorem 2.1.2. We build on Misiurewicz's proof of the variational principle for topological entropy as found in [Wal82]. Basically, we need to augment it with arguments that keep track of the averages of the observable. This is trivial for the proof of (i). Then the natural measures (i.e. limits of those equidistributed on the appropriate (; n)-separated sets) have the right averages, and we just quote [Wal82] claiming that their entropy is as required. 41 Part (ii) is more involved; we are forced to interweave the proof from [Wal82] with several arguments including application of Egoro's and Birkho's theorems. Proof of (i). We will show that, for any > 0, there is 2 M(X; f ) with R d 2 V and h (f ) > hV (f; ) . Choose > 0 so that hV (f; ; ) > top hVtop (f; ) . From the denition of hV (f; ; ), there exists a sequence nk 2 N, nk ! 1 and (; nk )-separated sets Ek such that limk!1(1=nk ) ln #Ek = hV (f; ; ) and (1=nk )Snk (f; )(x) 2 V , for x 2 Ek . Consider the uniform k 1 f j. atomic measures on Ek , call them Ek , and set k := (1=nk ) Pnj=0 Ek R R Note that, by convexity and closedness of V , dk = (1=nk )Snk (f; ) dEk 2 V . Passing perhaps to the subsequence of (nk ), we can assure that k 's weak R converge to some probability measure . Clearly d 2 V . More importantly, Misiurewicz proves that h (f ) limk!1 1=nk ln #Ek (see part (2) of the proof of Th.8.6., p. 189 in [Wal82]), so h(f ) hV (f; ; ) > hVtop (f; ) . Q.E.D. Proof of (ii). We will show that: if hUtop (f; ) > 2 and 2 M(X; f ) is such that Z nlim !1(1=n)Sn (f; )(x) = := d 2 U; (2:7) for -almost all x in X , then h (f ) hUtop (f; ) + ln 2 + 1. Clearly, (2.7) is R satised by ergodic with d 2 U ; however, unlike ergodicity, (2.7) is invariant under replacing f and by the iterate f m and by (1=m)Sm(f; ) respectively. By considering arbitrarily high iterates and using the power rule (Proposition 2.1.3), we conclude that h (f ) hUtop (f; ). 42 Let A = A1; :::; Ak be an arbitrary nite Borel partition of X . Choose > 0 so that < 1=(k ln k). Using Egoro's theorem, we may nd compact sets Bj Aj , with (Aj nBj ) < , 1 j k, such that (1=n)Sn(f; ) converges uniformly to on every Bj . Now, because of this uniform convergence and openness of U and since 2 U , there is 0 < d < 1 such that (1=m)Sm(f; )(x) + 2d f(x) : x 2 X g U; (2:8) for any x 2 Skj=1 Bj , and suciently large m 2 N. Put B0 := X n SfBj : 1 j kg and introduce a partition B := fB0; B1 ; :::; Bk g. Now, for 1 j k, we have Bj Aj , so the entropy H(A=Bj ) of A restricted to Bj vanishes for those j 's. Also, by the trivial estimate (see Corollary 4.2.1 p. 80 in [Wal82]), H (A=B0) ln k. Thus we get conditional entropy H(A=B) = P0jk (Bj )H (A=Bj ) (B0) ln k k ln k < 1. Consequently, (by (iv) of Th.4.12, p. 89 in [Wal82]) we have h (f; A) h (f; B) + H (A=B) < h(f; B) + 1: (2:9) For each n, let us split Bn := Wjn=01 f j B into B1n := fB 2 Bn : B B0 \ ::: \ f bdnc B0g and B2n := Bn n B1n. Note that #B1n (#B)n bdnc , thus, for suciently large n, we have #B2n #Bn (#B)n bdnc (1=2)#Bn: (2:10) Put Kn := fx 2 X : (1=m)Sm(f; )(x) 2 U for all m ng. We claim that the union S B2n is contained in Kn for suciently large n. Indeed, if x 2 A 2 B2n, then there is s, 0 s bdnc; such that f s(x) 2 Bj with j 6= 0. 43 Now, a simple estimate gives k(1=m)Sm(f; )(x) (1=m)Sm(f; )(f s(x))k 2bdnc sup k(z)k=m 2d zsup k(z)k; z2X 2X so, by the choice of d (see (2.8)), we have (1=m)Sm(f; )(x) 2 U . Adopting Misiurewicz's trick, consider C := fB0 [ B1 ; :::; B0 [ Bk g. This is an open cover of Kn (also X ). Write N (C n ; Kn) for the minimal cardinality of a subfamily of C n := Wjn=01 f j C that covers Kn, and let Cn be such a subfamily with #Cn = N (C n ; Kn). By our claim, each B 2 B2n lies in Kn; therefore, there exists C 2 Cn containing B . It is also clear from the denition of C that at most 2n elements of B2n can t into any xed element of Cn . We conclude that #B2n N (C n; Kn)2n: (2:11) By Th. 7.7 in [Wal82], if is the Lebesgue number of C , then N (C n ; Kn) s(=2; n; Kn), (where s(=2; n; Kn) is the maximal cardinality of an (=2; n)separated subset of Kn). Note that every (=2; n)-separated subset of Kn has the averages (1=n)Sn(f; )(x) in U , for large n. Thus, for those n, we get N (C n ; Kn) sU (f; ; =2; n). Now, combining (2.10), (2.11) and the last inequality, we get, for large n, H(Bn) ln #Bn ln(2#B2n) ln 2n+1sU (f; ; =2; n) : Passing to the limit, rst with n ! 1 then with ! 0, we obtain h (f; B) ln 2 + hUtop (f; ): 44 From (2:9), h (f; A) ln 2 + hUtop (f ) + 1, and by arbitrariness of A we get h(f ) ln 2 + hUtop (f; ) + 1, as promised. Q.E.D. 2.2 Topological entropy at the asymptotic average for subshifts of nite type (v) As an example of utilizing Theorem 2.1.1 to calculate htop (f; ), we consider the case of a transitive subshift of nite type with locally constant observable. This setting is exactly what we will need to analyze the case of pseudo-Anosov maps in the next chapter. t) (f; ) In the rst subsection, using standard techniques, we identify p(top with the leading positive eigenvalue of the appropriate transfer matrix. This (v) (v) gives an explicit real analytic formula for htop (f; ). We also note that htop (f; ) is realized as the metric entropy for the corresponding Gibbs state. The second subsection is devoted to showing that, apart from having the associated ergodic Gibbs state, each v 2 ri((f; )) is the asymptotic average for a compact invariant set with the topological entropy arbitrarily close to (v) htop (f; ). In most of our formulations we restrict attention to v in ri((; )) since ultimately (in the case of torus homeomorphisms) only such rotation vectors will be considered. However, the case of v in the boundary of (f; ) can be reduced to that of v 2 ri((; )) by passing to a smaller subshift of nite type using the scaold construction (taken from [MT91]). We briey describe this 45 (v) trick in the last subsection and show how it relates to htop (f; ). (v) For subshifts of nite type, the entropy htop (f; ) contains much more rened information that what is suggested by its denition. It can actually be used to count the number of periodic orbits with a prescribed average of the observable ([MT91]). This belongs to a topic that parallels the classical theorems on the distribution of the prime numbers via the formalism of Zeta functions. There is a large amount of published material on the subject, see for example [PP90] and the references within. 2.2.1 Transfer matrix, Gibbs states, and explicit formulas Consider a transitive subshift of nite type : ! with a locally constant observable function : ! Rd. Without any loss of generality, we can assume that there is an irreducible ([Wal82]) binary matrix A = (ai;j )1i;jN and a Rd-valued matrix ( i;j )1i;jN such that = fx 2 f1; :::; N gZ : axi ;xi = 1; i 2 Zg and (x) = x ;x for x 2 . The metric d on , that we shall use, is given by d((xk ); (yk )) := 2 jkj, where k is the closest to 0th position at which xk and yk dier. +1 0 1 A convenient representation of this setting is achieved by interpreting points of as bi-innite paths in the transition graph G f1; :::; N g2 with vertices f1; :::; N g and an edge (i; j ) from i to j whenever ai;j = 1. By a path in G we understand a map : fk; :::; l 1g ! G, where k; l 2 Z[f 1; +1g, and (i + 1) emanates from the endpoint vertex of (i), k i < l 1. The 46 number l k is the length of the path, denoted by l(). A loop is a path with the endpoint of (l 1) coinciding with the starting point of (k). The loops visiting any vertex of G at most once are called elementary. Clearly, there is a nite number of the elementary loops in G. Each periodic orbit of determines uniquely a loop with the length equal to the period. To keep track of the averages of , each edge (i; j ) 2 G is tagged with the vector weight P i;j . By we denote the sum of the weights along the edges of the path . If is a loop, we write () for the average (1=l()) P . Transitivity of guarantees that any pair of vertices i; j 2 G can be connected by a path i;j in G. We will refer to the paths in a xed collection fi;j g(i;j)2G as connecting paths and call t := maxfl(i;j ) : i; j g a connecting length of G. The shift map is expansive with the expansiveness constant = 1 (see [Wal82]); therefore, in an analogous way to the usual topological entropy, one (v) can easily verify that htop (; ) can be calculated on (1; n)-separated sets. If S is (1; n)-separated, then it determines a collections of #S dierent paths in G of length n | for a point (xi ) 2 S , the corresponding path passes through x0 ; :::; xn and Sn(; )(x) = P . Clearly, every collection of dierent paths of length n arises from a (1; n)-separated set in this way. Thus we can write (v) htop (; ) = rlim lim sup(1=n) ln sBr (v) (; ; 1; n) = !0 n!1 lim lim sup(1=n) ln #f : path in G; l() = n; (1=n) r!0 n!1 X (2:12) 2 Br (v)g: Since every path can be extended to a loop by adding an appropriate 47 connecting path, we have (v) htop (; ) = rlim lim sup(1=n) ln #f : loop in G; l() = n; () 2 Br (v)g: !0 n!1 (2:13) One can even insist that all the loops above depart from an arbitrarily xed vertex. The \lim sup" in (2.13) can not be replaced with \lim inf" unless is mixing. This (slight) inconvenience is oset by the following simple fact (cf. the example in Appendix A). Fact 2.2.1 In the formula (2.12), one can replace lim sup with lim inf. Proof. Take an arbitrary 0 < < 1. We will show that with \lim inf" we get (v) at least htop (; ). Fix v 2 Rd and r > 0. By (2.13), there exist arbitrarily large n0 2 N such that there is a collection of dierent loops of length n0 (v) with cardinality # exp (htop (; )n0 ) and (1=n0) P 2 Br=2(v), 2 . As noted above, we can assume that all these loops depart from a xed vertex of G so that we can concatenate them. For any n 2 N, set k := bn=n0 c. By considering all possible (# )k concatenations of k-tuples from k , we conclude that (v) sBr (v) (; ; 1; n) sBr= (v) (; ; 1; kn0 ) (# )k exp (htop (; )kn0 ) 2 (where for the rst inequality n must be suciently large). Taking the logarithms and \lim inf" with n ! 1 nishes the proof. Q.E.D. Combination of the results in the preceding section of this chapter with the known facts about subshifts of nite type gives us the following theorem 48 | very much in the spirit of [Bow75, Rue78]. (A reader not familiar with the Gibbs states may want to consult [Bow75].) Theorem 2.2.1 Suppose that (as above) : ! is a transitive subshift of nite type with observable : ! Rd that depends only on the rst two symbols. If A = (ai;j )1i;jN is the corresponding binary matrix and G is the corresponding transition graph with weights ( i;j )1i;j N , then: (i) (; ) is a convex polyhedron with vertices in f() : is an elementary loop in Gg; (v) (ii) htop (; ) is convex continuous on (; ) and strictly convex and real analytic on ri((; )); t) (; ) = ln((t)), where (t) is the leading positive eigenvalue of the (iii) p(top transfer matrix A(t) = (ai;j expht; i;j i)1i;j N , t 2 Rd ; (v) (v) (iv) htop (; ) = p^top (; ), for v 2 ri((; )); (v) (v) htop (; ) = hverg (; ) = hvmeas(; ) = h v (), where v 2 ri((; )) ( ) and (v) is the Gibbs measure corresponding to the function on given by t) (; ) (where r x 7! ht; (x)i, with t 2 Rd uniquely determined by v = rt p(top t is the gradient with respect to t). Remark. The dimension of (; ) may be lower than d. Thus real analyticity in (ii) is with respect to the ane hull of (; ). Nevertheless, even though we have chosen not to do so, one can always assume that (; ) has the dimension of the target space of . This is achieved by replacing with its projection onto the ane hull of (; ). 49 Remark. Let us add to (v) that, since is locally constant, the Gibbs measure is a Markov measure. More specically, if u = (ui)i=1;:::N and v = (vi)i=1;:::N are the eigenvectors, corresponding to the eigenvalue (t), of A(t) and of the transpose A(t) respectively, then this Markov measure is determined by a unique stochastic vector proportional to (uivi )i=1;:::N and by the stochastic matrix ( 1A(t)i;j vj =vi)i;j=1;:::N (cf. p. 194 in [Wal82].) Also, ui; vi > 0, i = 1; :::; N , so this measure is supported on the whole . Remark. For a subshift of nite type that is not transitive, the entropy (v) htop (; ) is the maximum of the entropies on the transitive components. The terminology transfer matrix comes from statistical mechanics (see [Ell85, Rue78]). Part (iii) of Theorem 2.2.1 is the essence of the transfer matrix method (see [May91] for an exposition). Proof of (i). This has been apparently realized by many; it is implicitly con- tained in [Fri82b], mentioned in [Kwa92], and proved in the full generality in [Zie95]. One way to argue is as follows. By transitivity, the rotation set is the closure of the -averages ( ) taken over all loops in G. Thus it is enough to see that, for every loop , the sum P can be represented as a sum of P 's for a number of elementary loops . This can be done by induction with respect to the length of . If is not elementary, then certain vertex v repeats at least twice and, by taking the two segments of between two dierent occurrences of v, one can \pinch" to two shorter loops without changing the sum of weights. Q.E.D. 50 Proof of (iii). The topological pressure is given by t) (; p(top ) = nlim !1(1=n) ln X( )! D X E exp t; : path in G; l() = n ; (2:14) or, in terms of loops, we can write (cf. (2.13) and [May91]) t) (; p(top ) = lim sup(1=n) ln n!1 X( )! D X E exp t; : loop in G; l() = n : (The convergence takes place for mixing .) The right-hand side can be written as lim sup(1=n) ln(trace(A(t)n)) = ln((t)); n!1 where the matrix A(t) = (ai;j expht; i;j i)1i;jN , and (t) is its leading positive eigenvalue given by the Perron-Frobenius theorem ([Wal82]). Q.E.D. Proof of (iv). Because A(t) is a non-negative irreducible matrix, the lead- ing positive eigenvalue (t) is a simple root of the corresponding characteristic polynomial, and so it depends real-analytically on t 2 Rd. In this way t) (; ) is smooth; and consequently, its convex conjugate is strictly convex p(top on the interior of the eective domain (see p. 253, [Roc72], cf. (2.15) below). We are done by Theorem 2.1.1. ( The eective domain is equal to (; ) by Corollary 2.1.1.) Q.E.D. t) (; ) is smooth, its convex conjugate p^(v) (; ) Proof of (v). Since by (iii) p(top top (v) can be calculated for v 2 ri dom p^top (; ) [Roc72]) as follows (cf. Th. 23.5, (v) t) (; ); p^top (; )) = hv; ti p(top (2:15) 51 (t) where t is uniquely determined by rt ptop (; ) = v. Moreover, we observed (v) while proving (iv) that (; ) = dom(^ptop (; )). Let (v) be the Gibbs state for ht; i. According to Ruelle (see Th.1.22 in [Bow75]) it is an (ergodic) equilibrium state, that is t) (; p(top E D Z ) = h v () + t; d(v) : ( ) The ane function : t0 7! h v () + ht0; ( ) R d(v) i is majorized by the convex t) (; ) due to the variational principle, and, by the above equality, function p(top t) (; ). Comparison with (2.15) yields R d(v) = v it is actually tangent to p(top (v) and h v () = p^top (; ). We are done by (iv) and Corollary 2.1.3. Q.E.D. ( ) (v) Proof of (ii). While proving (iv) we observed strict convexity of p^top (; ) = (v) t) (; ), htop (; ), for v 2 ri((; )). By (2.15) and real-analyticity of p(top (v) (; ) is real-analytic on ri((; )). v 7! htop Still, one needs to show convexity on the whole (; ). Then continuity (v) (; ) is lower semi-continuous by (iv) of Proposition follows; indeed, htop 2.1.1, and, by Th. 10.2, p.84 [Roc72], it is upper semi-continuous as a convex function on a polyhedron. To demonstrate convexity, it is enough to show the following: if v1 ; v2 2 (; ) and v = v1=2 + v2 =2, then (v) (; ) (1=2)h(v ) (; ) + (1=2)h(v ) (; ): htop top top 1 2 It is routine to extend this inequality to 2-adic convex combinations and then (v) (by upper semi-continuity of htop (; )) to arbitrary convex combinations. 52 Fix an arbitrary r > 0. By (2.12) and Fact 2.2.1, there are arbitrarily large n0 2 N for which one can nd two families 1 and 2 of paths 1 in G, (vi ) with length n0, so that # i exp (htop (; ) r)n0 and P 2 n0 Br (vi) for 2 i, i = 1; 2. Moreover, we may assume that n0r exceeds the connecting length t. For each pair of paths (1; 2) in 1 2 we can nd a connecting path so that := 12 is a path in G. At least # 1# 2 =t of thus obtained paths will have the same length n := l( ) = 2n0 + l (1 + r)2n0; where l := l( ). Let then (2:16) be a collection of those 's. Observe that, if 2 , (1=n) X 2 Br+ (v); where accounts for the contribution introduced by and is smaller than r for large n0 . Thus we have v) v) sB r (; ; 1; n) # exp (h(top (; ) r)n0 exp (h(top (; ) r)n0 =t: 2 1 2 By passing with n0 to innity, while keeping in mind (2.16), we get hBtopr (v) (; ) (2(1 + r)) 2 1 (v ) v ) (; ) 2r : htop (; ) + h(top 1 2 We are done by the arbitrariness of r. 1 These can be actually chosen to be all loops through a xed vertex. Q.E.D. 53 2.2.2 Observable shadowing with the right entropy Let , , , and G be as in the preceding subsection. By (v) of Theorem 2.2.1, we know that, for any v in ri((; )), there is an ergodic measure with (v) the metric entropy equal to htop (; ). That measure is supported on all of (see the remark after Theorem 2.2.1), and the support could not possibly be smaller as shown by the following proposition. Proposition 2.2.1 Suppose that v 2 ri((; )). If K is a compact -invariant (v) (v) (jK ) < htop (; ). proper subset of with ( jK ; ) = v, then htop The proposition also demonstrates that the following theorem is optimal. Theorem 2.2.2 For v 2 ri((; )) and 0 < < 1, there exists a compact (v) -invariant set K such that htop (jK ) htop (; ) and (jK ; ) = v. In fact, K is such that, for some constant C , kSn(; )(x) nv k C , x 2 K , n 2 N. This theorem, for the special case of with values in Z and v 2 Z, is proved in [MT92]. Also, in [Zie95] it is shown how to nd for each v 2 (; ) a compact invariant set K with (jK ; ) = v. Our methods, developed independently of [MT92, Zie95], are dierent. Let us also note that, using the scaold construction of the next subsection, one can get a stronger result for hverg , see Theorem 2.2.4. The proposition is simple, so we prove it rst. We need the following lemma. 54 Lemma 2.2.1 If K is a compact, -invariant proper subset of , then, for (t) t) (; ). any t 2 Rd , we have ptop (jK ; ) < p(top Proof of Proposition 2.2.1 from Lemma 2.2.1. Pick v 2 ri((; )). There is (v) (t ) the corresponding t0 2 Rd such that htop (; ) = hv; t0i ptop (; ), (see 0 (2.15)). Using Theorem 2.1.1 for the rst inequality and the lemma for the third, we obtain (v) t) ( j ; )g htop (jK ; ) sup fhv; ti p(top K t2R (t ) (v) (t ) hv; t0i ptop (jK ; ) > hv; t0i ptop (; ) = htop (; ): 2 0 0 Q.E.D. Proof of Lemma 2.2.1. Let be the Gibbs measure associated with ht; i. Its characteristic property is that there is a constant C > 0 such that, for any x = (xi)i2Z 2 and n 2 N, ([Bow75]) t) (; ) C; C 1 ([x]n0 )= exp hSn(; )(x); ti np(top (2:17) where [x]n0 := fy 2 : yi = xi ; i = 0; :::; n 1g is a n-cylinder. Since K is proper and is ergodic and supported on the whole , we have (K ) = 0. We can contain K in a union of disjoint cylinders that total to arbitrarily small measure; more concretely, for suciently high n0 2 N, there exists a discrete subset En of such that K Sx2En [x]n0 and (Sx2En [x]n0 ) (10C ) 1. Now, for n 2 n0 N, consider any (1; n)-separated subset of S K . Any n0cylinder that is determined by n0 consecutive symbols of a point in S must be 0 0 0 0 0 55 one of [x]n0 ; x 2 En . This, together with (2.17), yields the following estimate 0 0 X n=n Y0 X i=1 x2En0 x2S exphSn(; )(x); ti n=n Y0 X i=1 x2En0 exphSn (; )(x); ti 0 t) t) n=n C([x]n0 ) exp np(top (; ) C (10C ) 1 exp n0p(top (; ) : 0 0 Taking the logarithms and passing to the limit with n ! 1, we get (by (2.14)) t) ( j ; ) p(t) (; ) + (1=n ) ln(1=10): p(top K 0 top Q.E.D. We turn our attention to Theorem 2.2.2 now. As most of proofs in this section our argument hinges on concatenation of paths in G, however this time we need an original approach to control the averages. Sacricing the eciency of exposition, we take an opportunity to present a general technique that easily yields trajectories in that have an a priori prescribed behavior of the averages. (See Lemma 2.2.2, ahead.) We start with a simple characterization of convex polyhedra that may be interesting by itself. (Below, for two sets A; B Rd, A + B stands for the Minkowski sum, i.e. A + B = fa + b : a 2 A; b 2 B g.) Theorem 2.2.3 (characterization of convex polyhedra) For a compact convex 0 Rd , the following are equivalent : (i) 0 is a polyhedron; S (ii) there exist m 2 N and !1 ; :::; !m 2 0 such that 0 + 0 = mi=1 !i + 0 : Moreover, the !i's can be chosen so that they are convex combinations with rational coecients of the extremal points of 0 . 56 Proof of Theorem 2.2.3. To prove that (ii) implies (i), we just notice that, for x; y 2 0 : if the sum x + y is an extremal point of 0 + 0 = 2 0 , then both x and y must be extremal and equal to each other. Thus m bounds from above the number of extremal points of 0 what makes it a polyhedron. To prove (ii) from (i), we rst note that ane transformations of 0 do not aect validity of (ii). Since every polygon with n vertices is an ane image of the standard simplex n = fx 2 Rn : xi 0; P xi = 1g Rn, we may consider only the case when 0 = n. We claim that [ 2 n = fw + n : w = (w1; :::; wn) 2 n ; wi 2 (1=n)N; i = 1; :::; ng: To get this, x any x = (x1 ; :::; xn) 2 2 n . Clearly Pni=1 xi = 2, so we see that Pni=1bnxi c Pni=1(nxi 1) = 2n n = n: Hence, we can choose ki 2 N, 0 ki bnxi c; i = 1; :::; n, such that Pni=1 ki = n. Now, by setting wi := ki=n and yi := xi wi; i = 1; :::; n, we get Pni=1 wi = n=n = 1 and Pn y = 2 1 = 1, so both w = (w ) and y = (y ) belong to n. Clearly i i i=1 i x = w + y. Q.E.D. The following \shadowing type result" is the consequence of Theorem 2.2.3 that we are interested in. Corollary 2.2.1 Suppose that 0 and !i's are as in Theorem 2.2.3. If x : R+ ! Rd is piecewise C 1 with x(0) = 0 and x0 (t) 2 0 , t 2 R+, then there exists a sequence 2 f1; ::; mgN such that x(n) 2 ! + ::: + !n + 0 , for all n 2 N. 1 1 57 Proof. We proceed by induction with respect to n. Since x0 (s) 2 0 , by R convexity of 0 , x(1) = 01 x0 (s)ds 2 0 taking care of the case n = 1. For R the induction step, write x(n + 1) = x(n) + nn+1 x0 (s)ds 2 x(n) + 0 and use induction hypothesis to see that x(n +1) 2 ! + ::: + !n + 0 + 0 . By (ii) of Theorem 2.2.3, one can to nd 1 n m so that x(n+1) 2 ! +:::+!n +0 . Q.E.D. 1 1 1 Let us now explain how Corollary 2.2.1 renders orbits in (or equivalently paths in the graph G) with prescribed behavior of the Birkho averages of . By transitivity, we can nd loops 1, ... r , all emanating from a common arbitrarily chosen vertex i0 , such that 0 := convf(1 ); :::; (r )g approximates (; ) as well as we wish. Now, let !i, i = 1; :::; m, be as in (ii) of Theorem 2.2.3. Since !i's are all rational convex combinations of the vertices, by taking appropriate concatenations of i's, we can obtain loops 1; :::; m so that !i := (i), i = 1; :::; m. Moreover, replacing each loop i with its appropriate multiple we can make all i's of equal length l0. One can concatenate i's at will: if q 2 N and 2 f1; :::; mgq , then := :::q is a loop in G. 1 The following lemma is in fact much stronger than what we need for Theorem 2.2.2. For a path in G, by mn we understand its restriction to fm; :::; n 1g (if it is well dened). Lemma 2.2.2 If x : R ! Rd is piecewise C 1 with x(0) = 0 and x0(t) 2 0, t 2 R, then there exists a bi-innite path in G following x, i.e. X n m (x(n) x(m)) A := 4l0 diam(0 ) + 2l0 sup k (x)k; x2 58 for all n < m; n; m 2 Z. Moreover, for n 2 l0 N, the edge (n) ends at the vertex i0 . Clearly, the smoothness hypothesis is supercial. Proof. By applying Corollary 2.2.1 to the rescaled path t 7! (1=l0 )x(l0 t) twice, once with time running forward (t > 0), once with time running backward (t < 0), we get 2 f1; :::; M gZ such that for any k; l 2 Z with k < l, X l 1 ! (1=l0)(x(l0 l) x(l0 k)) 2 diam(0 ): i =k i Take for the innite concatenation := = ::: ::: : By multiplying the above inequality by l0, while remembering that !i = (i ) = (1=l0) Pi , we see that 1 0 1 (x(ll0 ) x(kl0 )) 2l0diam(0): The above inequality for restrictions of to intervals between multiples of l0 yields the result because, for m; n with jm nj l0 , both k Pmn k and kx(n) x(m)k are trivially bounded by l0 supx2 k (x)k and l0 diam(0) respectively. X kl ll 0 0 Q.E.D. Proof of Theorem 2.2.2. We will construct a sequence of natural numbers ni ! 1 and a sequence of (1; ni)-separated sets Ei such that and, for y 2 Ei, (v) #Ei exp(htop (; )ni) (2:18) kSk (; )(y) kvk C; (2:19) 59 where k 2 N and C is a constant independent on i. Then the set K := cl(Sl2N;i2N l (Ei)) satises the assertion of Theorem 2.2.2. This is because (2.19) is satised on a closed set of y 2 and, if it holds for y, then it holds for (y); 2(y); ::: with C replaced by 2C . Fix v 2 ri((; )). In constructing Ei we will think of its points as innite paths in G. As in the discussion preceding Lemma 2.2.2, choose loops 1 , ...m so that v 2 ri(0 ), where 0 = convf(1); :::; (m )g. Recall that all i's have a common starting vertex i0 . Let > 0 be rational and such that 1=(1 + ) p. Take r > 0 satisfying v B2r= (0) 0 . From the denition (v) of htop (; ) rephrased as (2.13), nd n0 2 N and a collection S0 of loops in (v) G, with length n0 and starting at i0, such that #S0 exp(htop (; )pn0 ) and (1=n0) P 2 Br (v) for any 2 S0. Require also that n0 is large enough so that A = 4l0 diam(0) + 2l0 sup k (x)k rn0: x2 (2:20) With no loss of generality one can assume that n0 2 l0 N. Set ni := in0 (1 + ), i = 1; 2; ::: . Fix i 2 N, and choose 1; :::i 2 S0 arbitrarily. With this data we will associate a path of length ni and of the form 1122::::ii , where j 's will be chosen so that X 1 1 :::j j nj v rn0; j = 1; 2; :::; i: (2:21) 60 P We dene j 's inductively. We start with nding 1. Observe that, since 2 n0 v + Bn r (0), 1 0 n0 (1 + )v X !. (n0 ) 2 (n0 v + Bn r (0))=(n0) = v + Br= (0) 0: 0 1 Consequently, there is a (linear) path x(t) 2 Rd; 0 t n0 , such that x0 (t) 2 0 ; t 2 [0; n0], x(0) = 0, x(n0 ) = n0(1 + )v P . From Lemma 2.2.2, we get a path 1 such that l(1) = n0 and 1 X x(n0 ) A rn0 : 1 Note that 1 is in fact a loop that starts (and ends) at the vertex i0 , and so is 1; thus we can concatenate the two. By the denition of x, P +x(n0 ) n1 v = 0, so from the above inequality we verify (2.21) for j = 1: 1 X 1 1 X X + x(n0 ) n1 v + n1 v 1 1 x(n0 ) rn0: (2:22) To get j+1, having 1; :::; j already dened and satisfying (2.21), we repeat the above procedure with a tiny modication. Observe that, since P 2 n0 v + Bn r (0), using (2.21) as the induction hypothesis, we get j +1 0 0 @nj+1v 1 . A (n0 ) = :::j j j 0 1 . X X @n0 v + (n0 v ) + (nj v )A (n0) X 1 1 2 j+1 +1 1 1 2 :::j j 2 (n0v + B2n r (0))=(n0) = v B2r= (0) 0 : 0 As before, we conclude that there is a (linear) path x(t) 2 Rd; 0 t n0 , such that x0 (t) 2 0 ; t 2 [0; n0 ], x(0) = 0, x(n0 ) = nj+1v P :::j j . 1 1 +1 61 From Lemma 2.2.2, we get a path j+1 such that l(j+1) = n0 and X j+1 x(n0 ) A rn0: For the concatenation 11:::j+1j+1 we have nj+1v ::::j j X X + x(n0 ) nj+1v + j ::::j 1 1 X 1 1 +1 +1 +1 +1 x(n0 ) 0 + rn0: This ends the induction step. The above procedure yields (#S0)i dierent paths of the form = 11:::ii, because we have that many choices of 1; :::; i 2 S0. To get from each such a point of Ei , just extend the path = 11:::ii to a bi-innite path. In this way, Ei is indeed a (1; ni)-separated set with i (v) (v) (; )nip=(1 + ) : #Ei = (#S0 )i exp(htop (; )pn0) = exp htop By the choice of , (2.18) follows. The (2.19), for k ni, is an immediate consequence of (2.21) (with C depending on n0 ). To get it right for all k, one just has to exercise some care while extending 's to the bi-innite paths. Namely, the semi-innite paths that are concatenated to must follow x(t) := tv, as in Lemma 2.2.2. Q.E.D. 2.2.3 The scaold of subshifts of nite type The following discussion (based on [MT91]) explains how to use the re(v) sults of the previous sections to learn about htop (; ) and hverg (; ) even if 62 v 62 ri((; )). Since (; ) is a convex polyhedron, we can think of (; ) as the total space of a polyhedral complex with strata (i) (; ), i = 1; 2; ::::d, dim((i) (; )) = i, where (d) (; ) = f(; )g and, inductively, (i 1) (; ) consists of all faces of polyhedrons in (i) (; ). (By a face of a k-dimensional convex polyhedron we understand a maximal (k 1)-dimensional convex polyhedron contained in its relative boundary.) Fix a point v 2 (; ). The support of v in (; ) is the unique polyhedron in the complex that contains v and is of least dimension among such, denote it by F. Observe that v 2 ri(F ). Let G(F ) be the graph obtained by the following pruning of G: an edge e of G is left in G(F ) , if and only if there is a loop in G passing through e that satises () 2 F . For the corresponding subshift of nite type we write (F ) : (F ) ! (F ). The collection of all thus obtained (F ) 's (as we vary v in (; )) together with the obvious inclusions between them forms what is called in [MT91] a scaold of subshifts of nite type. Note that subshifts (F ) need not to be transitive, but this adds only a trivial complication to the considerations (recall the remark after Theorem 2.2.1). A simple argument in [MT91] shows that any loop in G(F ) must have () 2 F . One can easily deduce then that ((F ) ; ) = F: It is also observed in [MT91] that any ergodic measure for which is supported on (F ); consequently, hverg (; ) = hverg ((F ) ; ); for v 2 F: R d 2 F (2:23) 63 As a corollary, we get the following extension of Theorem 2.2.2 to v 62 ri((; )). Theorem 2.2.4 Let v 2 Rd be such that hverg (; ) 0 (i.e. v = R d, for some ergodic probability measure ). For 0 < < 1, there exists a compact -invariant set K such that htop (jK ) hverg (; ) and (jK ; ) = v. In fact, K is such that, for some constant C , kSn(; )(x) nv k C , x 2 K , n 2 N. R d for an ergodic measure , then v belongs to (; ). Let F be the support of v in (; ). If F is a point, then one can take for K the whole subshift (F ) because, by (2.23), hverg (; ) = hverg ((F ); ). In general (v) (F ) v 2 ri(F ), so, by (v) of Th. 2.2.1, hverg (; ) = htop ( ; ), and we nd K (F ) by applying Theorem 2.2.2 to the subshift (F ) . Q.E.D. Proof. If v = (v) The behavior of htop upon passing from to the subshift (F ) is given by the following proposition. Proposition 2.2.2 For v 2 (; ) with support F (; ), we have (v) htop (; ) = convex hull (v) (F ) htop ( ; ) : (2:24) (If (F ) is transitive, then the convex hull is superuous.) We will give two proofs of the proposition. Short proof. Recall that is expansive making hvmeas (; ) upper semi-contin- (v) uous (cf. [Wal82]). Thus, since htop (; ) is convex, Corollary 2.1.2 yields (v) htop (; ) = hvmeas(; ) = convex hull hverg (; ) : 64 Using (2.23), we can write (v) (F ) (v) hverg (; ) = hverg ((F ) ; ) htop ( ; ) htop (; ): Taking the convex hull of each function above nishes the argument. Q.E.D. Note that ultimately the above proof owes its eciency to nontrivial results of ergodic theory. For a reader ready to fend o supercial technicalities, we supply the following argument unveiling elementary mechanism standing behind the proposition. (v) (v) (F ) Long elementary proof. Clearly htop (; ) htop ( ; ); and so, by convexity (v) (v) (v) (F ) of htop (; ), we have htop (; ) convex hull htop ( ; ) . We will prove the opposite inequality by showing that, for arbitrary 0 < < 1, (v) htop (; ) convex hull (v) (F ) htop ( ; ) : It is enough to consider the case when F is a face of (; ). Let V be the ane space spanned by F . Set ( d := min dist ((); V ) : X ) 26 V; elementary loop in G : (v) When htop (; ) = 0, there is nothing to prove. Otherwise, observe that one can nd > 0 such that, for suciently large n 2 N, (v) nN (n; n)#Gn exp (2n) exp (1 )htop (; )n ; (2:25) where N(k; m) is the binomial Newton symbol. Also, let W F be discrete and such that the balls fB (w)gw2W cover F = ((F ); ). 65 Pick 0 < r0 < and k0 2 N so that: if 0 is a path in G(F ) with l( 0 ) k0, then (1=l( 0)) P0 2 Sw2W Br (w), and, for w 2 W , 0 w) (F ) ( ; ) + )k ; for k > k0: sBr (w) ((F ); ; 1; k) exp (h(top 0 (2:26) (See the page 30 for denition of sBr (w)((F ); ; 1; k).) Now, take r > 0 small enough to have Nr(k0 + 1)=d ; (2:27) where N is the number of vertices in G. Consider for a while a xed loop in G of length l( ) = n and with ( ) = (1=l( )) P 2 Br (v). We will show that \most" of lies in G(F ). As described in the argument for (i) of Theorem 2.2.1, one can rearrange the edges of to get elementary loops 1; :::; p such that P = Ppi=1 Pi . Note that dist X ! X ; nV = i l(i)dist((i ); V ) #fi : (i) 62 V g: Since v 2 F V , P 2 Bnr (nv) implies that at most nr=d of i's have (i) 62 V . Since the length of an elementary loop is by denition less than N , at most Nnr=d of the edges in can belong to G n G(F ) . After deleting these edges, the path splits into at most nNr=d paths in G(F ), of which those longer than k0 are denoted by 1, ... q , q nNr=d, in the order of appearance along . The combined length of i's is greater than n nNr=d nNrk0 =d = n nNr(k0 + 1)=d (1 )n, where we used (2.27) for the last inequality. 66 One of the consequences is the following basic estimate X 0 (1=l( )) k (x)k: := 2 sup x2 i=1 i j =1 Furthermore, with each i associate wi 2 W chosen so that (1=l(i)) X wi r0 ; i = 1; :::q: i q q X X X l ( )) (1 = j This and (2.28) yield (1= q X j =1 l(j )) q X i=1 l(i)wi 2 Br++0 (v): (2:28) (2:29) (2:30) We claim that relatively few loops of length n as considered above can share the same i's (in the same order and positions along ) and also have a coinciding vector of wi's associated with i's. Indeed, by basic combinatorics, the number of possibilities for the choice of the deleted elements is bounded by n N(n; n)(#G)n and the number of choices of wi's by N(n; #W ) exp(n); where the right most inequality holds for large n. (v) By denition of htop (; ) (see (2.13)), there are arbitrarily large n 2 N (v) with at least exp((htop (; ) )n) dierent 's as considered above. The two combinatorial estimates and \the pigeon-hole principle" guarantee that some selection of wi's and the deleted elements (and thus also of i's) is shared by at least (v) (v) exp (htop (; ) )n =n N(n; n)(#G)n exp(n) exp htop (; )n 67 dierent 's, where the inequality is due to the choice of , see (2.25). For each i = 1; :::q, denote by Si a collection of all dierent i's showing in (v) those 's. The above estimate implies that #S1 :::#Sq exp htop (; )n . Also, by the choice of r0 and k0 and using (2.29), we get (wi ) (F ) #Si exp (htop ( ; ) + )l(i) ; i = 1; :::q: Consequently, after taking the logarithms and limsup with n ! 1, we get (v) (; htop 1 q 0 q X X (wi ) (F ) ( ; ) + ) @1= l(j )A l(i)htop j =1 i=1 and thus, if w := 1= Pqj=1 l(j ) Pqj=1 l(i)wi, then (v) htop (; ) convex hull w) ( (F ) ; ) h(top : By (2.30), when is shrunk to zero, w approaches v, and the desired inequality follows by lower semi-continuity of the right side above in the argument w. Q.E.D. Chapter 3 Topological entropy at the rotation vector for torus maps The rotation set, in the case of torus homeomorphisms, is very far from being a complete topological invariant; this makes sensible the search for ner invariants. While the presence of a particular vector v in the rotation set merely tells us that there is \some" dynamics with this asymptotic average displacement, one may try to measure the abundance of those dynamics. Realizing this philosophy, we associate with v the corresponding \topological entropy". We use quotation marks because, even though there is a well agreed on denition of the topological entropy of a map ([Wal82]), there are many ways of relativizing it to v. We will mainly use a simple topological concotion (v) htop and its measure theoretic counterpart hverg , both of which are extensively (v) discussed in Chapter 2 and Appendix A. For the pseudo-Anosov maps, htop and hverg coincide and are explicitly calculable once we know the Markov partition (Theorem 2.2.1). The pseudo-Anosov maps, our favorite examples, also (v) provide a way to give lower bounds for htop for a general map. 68 69 In the rst section, we dene the relevant entropies and state the lower bound they obey. In the second section, we translate the results of Section 2.2 to the case of the pseudo-Anosov maps. In the third section, combining the main theorems of Chapter 1 and Chapter 2, we prove the lower bound. Last (v) section is devoted to a concrete example of a calculation of htop . 3.1 Denitions and the main result Recall that for f 2 H(T2 ) and its lift F 2 H~ (T2), F id descends to the displacement function F : T2 ! R2. Note that (F ) = (f; F ). More generally, given an invariant set K T2 , we will write (F jK ) for (f jK ; F ). Denition 3.1.1 For v 2 R2, the topological entropy at the rotation vector v is (v) (F ) := h(v) (f; ) 2 f 1g [ [0; +1]: htop F top Similarly, we have hverg (F ) := hverg (f; F ) 2 f 1g [ [0; +1]: (v) Our ultimate goal in this chapter is to bound entropies hverg (F ) and htop (F ) from below in terms of the \size" of the rotation set and the relative distance of v to its boundary. To this end, we state the following denition. For a convex compact K R2 with int(K ) 6= ;, we dene the tent function of K by K (v) := inf fka vk=ka bk : v 2 convfa; bg; a; b 2 @K g ; v 2 int(K ): 70 Before we say more about the properties of K (v), we formulate our main result. (For the denitions of A() and I (), used below, see Section 1.1.) Theorem 3.1.1 (Main Lower Bound) There are universal constants C and C 0 such that, if f 2 H(T2 ) is a dieomorphism, F 2 H~ (T2) is its lift and v 2 int((F )), then (v) htop (F ) hverg (F ) maxfC ln+ A((F )); C 0=I ((F ))g (F ) (v): (3:1) Moreover, there exists an invariant compact set T2 such that (F j) = fvg and htop (f j) maxfC ln+ A((F )); C 0=I ((F ))g (F ) (v): (3:2) The proof of the theorem is given in Section 3.3; however, Section 3.2 is a necessary prerequisite for it. Let us now shed some light on K (v). First of all, it depends only on the ane properties of K . There is however a trivial bound K (v) dist(v; @K )=diam(K ): The \inmum" in the denition of K (v) is achieved when a and b are antipodal, meaning that there is a pair of parallel lines, one tangent to @K at a, and the other tangent to @K at b. Indeed, by compactness of @K and continuity of the norm, there are a and b such that v 2 convfa; bg and K (v) = ka vk=ka bk. Figure 3.1 below gives an idea of what would go wrong if a and b were not antipodal. 71 a v a2 a1 b2 b b1 Figure 3.1: When a and b are not antipodal. From Figure 3.1, note that ka1 vk < ka2 vk and kv b1 k > kv b2 k, so ka1 vk=kv b1 k < ka2 vk=kv b2 k = ka vk=kv bk. Thus ka1 vk=ka1 b1k < ka vk=ka bk, a contradiction. Finally, observe that K (v) is concave in v. In fact, we have ( K (v) = inf g(v) : g : K ! R+ ) concave with sup g(x) = 1 : x2K (3:3) (This is why we call K (v) the tent function of K .) For a proof, x v 2 K . If g : K ! R+ is concave with sup g = 1, then one can take a; b 2 @K such that convfa; bg contains v and a point w 2 K where g(w) = 1. By concavity, g(v) kkvw wbkk g(w) + kkwv bbkk 0 = ka vk=ka bk K (v): On the other hand, if K (v) = ka vk=ka bk, then there is a g such that g(v) ka vk=ka bk; simply consider the two parallel tangent lines through a and b and take g linear and constant on each of them. 72 3.2 The central example: pseudo-Anosov maps Throughout this section let f 2 H(T2) be a map that is pseudo-Anosov rel a nite set P and let F : R2 ! R2 be its lift to the universal cover. An abundance of rotation sets is exhibited by such examples. Indeed, we know that, given a convex polygon 0 with vertices in Q2, there is F 2 H~ (T2) such that (F ) = 0 ([Kwa93]). By Theorem 1.2.1 and the subsequent remark, we have G 2 H~ (T2) that is a lift of a pseudo-Anosov rel a nite set and has (G) = 0 . (v) (F ) and hv (F ) in the pseudo-Anosov case is Eective treatment of htop erg made possible by the method of symbolic dynamics. Proposition 3.2.1 ([FLP79]) There exists a Markov partition R = fR1 ; :::; RN g for f . If G is a graph with the set of vertices f1; :::; N g that has an edge from i to j whenever f (Ri ) \ int(Rj ) = 6 ;, then the subshift of nite type (; ) associated with G (see sec. 2.2) is mixing and factors onto f . More precisely, for any x = (xi)i2Z 2 the intersection Ti2Z f i(Rxi ) consists of a single point, denoted by p(x), and thus dened map p : ! T2 has the following properties: (i) p = f p, (ii) p is continuous and surjective, (iii) p is nite-to-one, (iv) p is 1-1 onto a topologically residual set consisting of all the points which full orbits never hit the boundary of the Markov partition. The above proposition gives a rather complete description of the dynamics of f in terms of the shift . To keep track of the averages of the observable 73 in the symbolic model, we need to associate with every edge (i; j ) of the graph G a vector weight i;j , reecting the displacement in the universal cover that is inicted on lifts of points in Ri that are mapped into Rj . One way to do that (taken from [Kwa92]) is as follows. Select a point zi in each Markov box Ri, i = 1; :::N . If (i; j ) 2 G, then f (Ri) \ int(Rj ) 6= ;, so there is a pair of corresponding lifts such that F (R~i) \ int(R~j ) 6= ;; dene i;j := z~i z~j ; where z~i 2 R~i and z~j 2 R~j are the lifts of zi and zj respectively. This denition does not depend on the choice of the lift R~i. Also, recall that the weights i;j can be thought of as an observable : ! R2 given by ((xi)i2Z) := x ;x . 0 1 Claim 3.2.1 For x = (xi )i2Z 2 and any n 2 N, we have kSn(f; F )(p(x)) Sn(; )(x)k 2 diam(R): In fact, F p and are cohomologous. Proof. Having xed a lift R~x of Rx , let R~ x be the lift of Rx such that z~1 z~0 = x ;x , let R~x be the lift of Rx such that z~2 z~1 = x ;x , etc. Thus dened sequence R~xj (j = 0; 1; 2:::), by the denition of , has the following 0 0 1 2 0 1 1 2 1 2 property: if q~ 2 R~x is a lift of q = p(x), then F i(~q) 2 R~xj , j = 0; 1; 2:::. The inequality follows because Sn(f; F )(q) = F n(~q) q~ and Sn(; )(x) = z~xn z~x , whereas z~j 2 R~xj , j 2 N. Moreover, if u(x) := q~ z~x , then F p = u u, making F p and cohomologous. Q.E.D. 0 0 0 Proposition 3.2.2 Suppose that F 2 H~ (T2 ) descends to f 2 H(T2 ) that is pseudo-Anosov rel a nite set. If (; ) is the corresponding subshift of nite 74 type and is the corresponding observable on , then (F ) = (f; F ) = (; ); and, for any v 2 int((F )), (v) (v) htop (F ) = hverg (F ) = htop (; ) = hverg (; ): (3:4) Moreover, (i) there is a fully supported ergodic probability measure (v) such that h v (f ) = R hverg (F ) and F d(v) = v, v 2 int((F )), (v) (ii) htop (F ) is strictly concave and real analytic in v on int((F )). ( ) Remark. In fact hverg (F ) = hverg (; ) for all v 2 (F ). Proof Proposition 3.2.2. Part (ii) follows from the main assertion by (ii) of Theorem 2.2.1. The statement about the rotation sets follows easily from (i) and (ii) of Proposition 3.2.1 coupled with Claim 3.2.1. To deal with the entropies, consider an invariant -ergodic probability measure . It pushes R R forward to := p( ), which is ergodic with respect to f ; and d = F d because p is cohomologous to F . Moreover, if is fully supported, then we claim that h (f ) = h (): (3:5) This is standard. Observe that is also fully supported; and, by ergodicity, the full measure is carried on the invariant residual set on which p is 1-1. Hence, the measure theoretical systems (T2; f; ) and (; ; ) are isomorphic via p. 75 Now x v 2 int((F )). Taking for the Gibbs state averaging to v as in (v) of Theorem 2.2.1, we see that (3.5) yields (v) (v) htop (f; F ) hverg (f; F ) h(f ) = h () = hverg (; ) = htop (; ): (v) (v) (f; F ) htop (; ) (see On the other hand, since f is a factor of , htop Proposition 2.1.4), so all the quantities above must be equal. This proves (3.4), and we see that (v) := is as stipulated by (i). Q.E.D. Proof of the remark. Since f is a factor of we have hverg (f; F ) hverg (; ) (see Proposition 2.1.4). To prove the opposite inequality, take a -ergodic R probability measure with d = v and set := p ( ). We need to show that h (f ) h (). Let E T2 be the full orbit of the union of the stable boundaries and the unstable boundaries of the Markov boxes, denoted by @ Rs and @ Ru respectively. If (E ) = 0, then h (f ) = h () (as in the proof of the proposition). Otherwise, by ergodicity, (E ) = 1. Since @ Rs is forward invariant under f and @ Ru is backward invariant under f , we have either (@ Rs ) = 1 or (@ Ru ) = 1. Say, we have the rst case (for the other use f 1). Then, h (f ) htop (f j@Rs ) = 0, where the vanishing of the topological entropy follows from the fact that @ Rs is a collection of smooth arcs (cf. [Bow78b]) with the length strictly contracted by f ([FLP79]). Since p is nite-to-one htop (jp (@Rs ) ) = 0 (see Th. 17 in [Bow71]); thus h () htop (jp (@Rs ) ) = 0. Our claim follows. Q.E.D. 1 1 Using the scaold construction (see the subsection 2.2.3 and Theorem 2.2.2), one can verify the following: for v 62 int((F )), if there are any measures 76 (v) realizing htop (; ), then they are the Gibbs states for the smaller subshift of nite type (F ) that corresponds to the support F of v in (F ). If is such a Gibbs state realizing hverg (; ), then the argument above shows that p ( ) realizes hverg (f; F ). As a corollary of Theorem 2.2.2 we get the following. Theorem 3.2.1 Suppose that f 2 H(T2) is pseudo-Anosov rel a nite set, F is its lift and v 2 int((F )). For any 0 < < 1, there is a compact set Q T2 (v) (F ). invariant under f such that (F jQ) = fvg and htop (f jQ) > herg Proof of Theorem 3.2.1. Take Q = p(K ), where K is as in Theorem 2.2.2. The fact that (F jQ) = (jK ; ) follows from Claim 3.2.1. The map p is nite-to-one, so we have htop (jK ) = htop (f jQ) (by Th. 17 in [Bow71] and the fact that entropy on a nite set must be zero). Q.E.D. Our focus on the case v 2 int((F )) is partially justied by the following fact. 1 Proposition 3.2.3 If f 2 H(T2) is pseudo-Anosov rel a nite set and F is its lift, then int((F )) = 6 ;. Proof of Proposition 3.2.3. Passing to an iterate of F and perhaps postcom- posing it with a deck transformation, we can assume that F has a xed point x = F (x) 2 R2. Fix a Markov partition for f . Denote by R the lift of the Markov box containing x. We will show that there are three non-collinear 1 This is essentially due to Fried, compare Theorem H in [Fri82a]. 77 vectors in (F ). Think of T22 := R2=(2Z)2, a 4-to-1 covering of T2 = R2=Z2 . The map F factors to f2 : T22 ! T22, which is also pseudo-Anosov rel a nite set. By transitivity of the unstable foliation ([FLP79]), the unstable leaf through the projection of x to T22 intersects the projections of int(R), int(R) + (1; 0), and int(R) + (0; 1). It follows that, for some large n 2 N, we have F n(R) \ int(R) + k0 6= ;, F n(R) \ int(R) + k1 + (0; 1) 6= ; and F n(R) \ int(R)+ k0 +(1; 0) 6= ;, where ki 2 (2Z)2 , i = 1; 2; 3. This puts k0=N , (k1 +(1; 0))=N and (k2 +(0; 1))=N in the rotation set of F . These three vectors are not collinear because the vectors k1 k0 + (1; 0) and k2 k0 + (0; 1) have the corresponding coordinates of opposite parity and so can not be linearly dependent. Q.E.D. 3.3 Proof of the main result We have f 2 H(T2 ) with a lift F 2 H~ (T2). Assume that int((F )) 6= ;; otherwise, there is nothing to prove. Fix v 2 int((F )) and take arbitrary > 0. One can choose a convex polygon 0 int((F )) with vertices in Q2 so that A(0) = A((F )), I (0 ) = I ((F )) and 0 (v) (F ) (v) . By Theorem 1.2.1, there is a nite set P (a union of primitive periodic orbits) and a map g 2 H(T2 ) such that: f and g are isotopic rel P , g is pseudo-Anosov rel P , g has a lift G 2 H~ (T2) with (G) 0 . The following theorem asserts that the dynamics of g are fully reected by f . It is a variation on Handel's global shadowing result in [Han85] that is most closely related to Theorem 3.2 in [Boy10]. (Also, this is the only place 78 where we are forced to use smoothness of f . One should be able to remove this obstacle.) Theorem 3.3.1 If f 2 H(T2) is a dieomorphism, P is a nite invariant set and g : T2 ! T2 is pseudo-Anosov rel P and isotopic to f rel P , then there is a compact invariant set Y T2 and a surjection h : Y ! T2 that is homotopic to the inclusion map Y ,! T2 and semi-conjugates f jY to g (i.e. h f jY = g h). The map g and the factor map h from the theorem can be lifted to G 2 H~ (T2 ) and an equivariant H : 1(Y ) ! R2 respectively, so that H F j (Y ) = G H . In particular, for x 2 Y , we have 1 kSn(f; F )(x) Sn(g; G h)(x)k D := 2 sup kH (y) yk; n 2 N: (3:6) y2 1 (Y ) Indeed, if x~ is a lift of x, the left side is equal to k(F n(~x) x~) (Gn H (~x) H (~x))k = kF n(~x) H F n(~x) + H (~x) x~k: W need the following easy corollaries, which proof we postpone. Corollary 3.3.1 Under the assumptions of Theorem 3.3.1, we have: (i) for any compact invariant Q T2, the set W := h 1 (Q) is compact invariant, (F jW ) = (GjQ) and htop (f jW ) htop (gjQ); (ii) for any 2 E (T2 ; g), there is 2 E (T2 ; f ), such that := h ( ), R d = R d and h (f ) h (g); G F (v) (v) (iii) for any v 2 Rd , htop (F ) htop (G) and hverg (F ) hverg (G); (iv) (G) (F ). 79 Proof of Theorem 3.1.1. To prove (3.1) of the theorem, observe that, by (ii) (v) of Theorem 2.2.1, htop (G) is a concave function of v; thus, from (3.3) and (iii) of Proposition 2.1.1, we have (v) x) (G) : x 2 R2 g (v ) = h (g ) (v ): htop (G) supfh(top top (G) (G) Using Theorem 1.1.1 and (iii) of Corollary 3.3.1, we obtain (v) (v) htop (F ) htop (G) maxfC ln+ A((G)); C 0=I ((G))g (G) (v); and the analogous inequality for hverg . Because 0 (G) (F ) and (G) (v) (F ) (v) (where is arbitrary), the inequality (3.1) follows. The assertion (3.2) is an immediate consequence of Theorem 3.2.1 via (i) of Corollary 3.3.1. Q.E.D. Proof of Corollary 3.3.1. Part (i): the equality of the rotation sets follows from (3.6); the entropy assertion is standard ([Wal82]). Part (iii): use (3.6) to note (v) that, in calculating htop (F ) and hverg (F ), one can use G h in the place of F and then recall the quotient rules | Proposition 2.1.2 and Proposition 2.1.4. The part (ii) can be extracted from the proof of Proposition 2.1.4. The part (iv) is immediate from (3.6). (It also follows from (iii) by (ii) of Proposition 2.1.1.) Q.E.D. Proof of Theorem 3.3.1. We will show how to reduce the result to Theorem 3.2 (and its proof) in [Boy10]. We start with a construction taken from [Bow78a] and rst used in a similar context in [LM91]. Let M be the compact surface with boundary obtained 80 by blowing up the points of P to small circular discs and then by removing its interiors. Denote by c : M ! T2 a continuous map that collapses the boundary circles of M to the corresponding points of P and is 1-1 elsewhere. Using the derivative of f at points of P , one constructs f1 : M ! M that factors to f via c, i.e. c f1 = f c. Now, let g1 : M ! M be Boyland's condensed homeomorphism in the isotopy class of f1 (see Case 5, p. 13 in [Boy10]). In our case, g1 is just a pseudo-Anosov map with some standardized behavior at the boundary | it is called (in [Boy10]) boundary-adjusted pA map. In particular, g1 factors to a map that is pseudo-Anosov rel P and (after perhaps some conjugation) coincides with g, i.e. c g1 = g c. Theorem 3.2 in [Boy10] yields a compact invariant set Y1 M and a surjective map h1 : Y ! M (homotopic to the inclusion) such that h1 f1 = g1 h1. The map h1 needs not to be continuous, however c h1 is. Put Y = c(Y1). Because the map f1 was obtained as a blow up of f , it is a tautology to say that it is uniformly continuous on int(M ) in the metric induced from T2 by c : int(M ) ! T2. Now, one has to trace word by word Boyland's proof of uniform continuity of c h1 on int(M ) while substituting in the place of the standard metric on int(M ) the one induced by c. As a result, we get uniform continuity of c h1 in this new metric. It follows that h := c h1 c 1 : Y ! T2 is a well dened and continuous map. This map has all the required properties. Q.E.D. 81 3.4 Example (v) We calculate htop numerically for a concrete example of a pseudo-Anosov map. This map is derived from a map in H(T2 ) via the method from Section 1.2 coupled with the Bestvina-Handel algorithm ([BH92]). Our discussion also (v) illustrates how results of Section 3.2 can directly bare on htop when dealing with a single explicitly dened map. Let H; V 2 H~ (T2) be given by H (x; y) := (x + j sin(y)j; y); V (x; y) := (x; y + j sin(x)j); and let h and v be the corresponding maps in H(T2 ). Consider the composition f := v h 2 H~ (T2) and its lift F := V H 2 H~ (T2 ). Note that F (0; 0) = (0; 0), F (1=2; 0) = (1=2; 1), F (0; 1=2) = (1; 1=2), and F (1=2; 1=2) = (3=2; 3=2). Thus the square Q := f(x; y) : 0 x; y 1g is contained in (F ). In fact, we have (F ) = Q because, by writing F = G h + H , one can verify that the range of the displacement function F sits in Q. Now, let P T2 consist of the three xed points of f that have (0; 0), (1=2; 0), (0; 1=2) for their lifts. The isotopy class of f rel P is represented by a map g that is pseudo-Anosov rel P (cf. Theorem 1.2.1). To nd out about g, we use the algorithm by Bestvina and Handel. The surface T2 n P deformation retracts to its spine consisting of the union of the oriented loops ; ; ; , marked on Figure 3.2. To record the isotopy classes of h; v; f , we postcompose them with the retraction and look at the 82 resulting transformation of the spine into itself. On Figure 3.2, the dotted lines indicate the images of and under the resulting action on the spine for h. Alternatively, we write h# : 7! , 7! , 7! , 7! , where the bar over an oriented path switches the orientation. y γ δ γ α β α β γ δ δ α α β β γ δ x Figure 3.2: The map h# on the spine. Similarly, for v, we get v# : 7! , 7! , 7! , 7! . For the composition f = v h, we get f# : 7! , 7! , 7! , 7! . It turns out that action of f# on the spine is not ecient; this means that, as we apply f# repeatedly to arcs making up the spine, we get paths that backtrack (see [BH92] for the denition). Bestvina and Handel describe moves that transform the map f# and the spine leading to an ecient map in a nite number of steps. The implementation of the algorithm, in our case, involves nine steps and is more tedious then instructive, so we skip it. The result is a graph contained in T2 n P (see Figure 3.3), which is a deformation retract of T2 n P , and a transformation g1 of this graph that is ecient. 83 c’ y h’ f’ c b’ g’ e’ f e a d h g b x Figure 3.3: The ecient map g1. The marked arcs are mapped by g1 as follows: a 7! a0 = a, b 7! b0 = b, c 7! c0 = ghbh, d 7! d0 = cfe, e 7! e0 = ef , f 7! f 0 = cfg, g 7! g0 = dae, h 7! h0 = fgh. As described in [BH92], the graph can be smoothed into a train track that carries a pseudo-Anosov map g which deforms to g1. The Markov boxes for g can be obtained by thickening the arcs c; d; e; f; g; h to rectangles. Actually, Figure 3.3 depicts not only g but a certain lift G of g. To determine the rotation set (G), we read o the gure the displacements under G: a 7! a b 7! b + ( 1; 1) c 7! g + ( 1; 1) h + ( 1; 1) b + ( 1; 1) h + ( 1; 1); d 7! c f e; 84 e 7! e f; f 7! c f + ( 1; 0) g + ( 1; 1); g 7! d + ( 1; 0) a + ( 1; 0) e + ( 1; 0); h 7! f + ( 1; 0) g + ( 1; 1) h + ( 1; 1): Tying this with the material in Section 2.2, we draw the weighted transition graph of the corresponding subshift of nite type | see Figure 3.4. (For clarity we skip weights (0; 0).) (-1,1) (-1,1) b a h (-1,0) (-1,1) c g (-1,1) (-1,0) (-1,0) (-1,1) (-1,0) f d e Figure 3.4: The transition graph. It is apparent from this graph that (by Proposition 3.2.2 and (i) of Proposition 2.2.1) (G) = convf(0; 0); ( 1; 0); ( 1; 1)g: Note that the map F (that we started with) is equivariantly isotopic rel 1 (P ) to G + (1; 0), not G. By Corollary 3.3.1, we have (v) (v+(1;0)) htop (F ) htop (G); v 2 R2 : 85 (v) To calculate htop (G), we extract from the graph its transfer matrix, see below. (The columns and rows, counted starting at the upper left corner, correspond to c, d, e, f , g, h. We skip a and b because they do not correspond to any Markov boxes.) 0 BB BB 0 BB BB BB 1 BB BB BB BB 0 A := B BB BB BB 1 BB BB BB 0 BB BB B@ 0 0 0 0 xy 2xy 0 1 1 0 0 0 1 1 0 0 0 0 x xy 0 x x 0 0 0 0 0 x xy xy 1 CC CC CC CC CC CC CC CC CC CC ; CC CC CC CC CC CC CC CC A (3:7) where x = exp( s) and y := exp t, (s; t) 2 R2. The characteristic polynomial of A is p(x; y; z) := det(A zI ) and can be found to be equal to p(x; y; z) = x4 y2 x3 yz x3y2z x4y2z + x2 yz2 + x3 yz2 + x3y2z2 6x2yz3 + xz4 + xyz4 + x2 yz4 z5 xz5 xyz5 + z6 : (u;v) Given a rotation vector (u; v) 2 int((G)), a way to nd htop (G) is through solving the following system of equations p(x; y; z) = 0; @p (x; y; z) = x @p (x; y; z); vz @z @x (3.8) (3.9) 86 @p (x; y; z) = uz @z @p (x; y; z): y @y (3.10) Provided the right solution (x; y; z) is taken, we have (v;u) htop (G) = ln(z xv y u): (3:11) s;t) = ln z , where z is the leading Indeed, by (iii) of Theorem 2.2.1, p(top positive root of the rst equation. By (iv) of Theorem 2.2.1, (v;u) (G) = supfsv + tu ln z : s; t 2 Rg; htop (3:12) where the supremum is achieved when v = @ (ln z)=@s = x(@z=@x)=z and u = @ (ln z)=@t = y(@z=@y)=z. This was used to obtain the two last equations of the system by taking partial derivatives of the rst one with respect to s, for one, and with respect to t, for the other. The system (3.8) may have many solutions. Continuation of a known solution is a way to select the right one. For example, one can start with x = 1 and y = 1 nding the leading positive root of p(1; 1; z) to be approximately 2:6103. Thus htop (g) = max h(v;u) (G) = ln 2:6103: (v;u) top The corresponding rotation vector, calculated from the measure of maximal entropy (see (v) of Theorem 2.2.1), is (vmax ; umax) = ( 2=3; 1=3): (v;u) Changing (u; v) away from (vmax ; umax), we may follow htop (G). In this way, using Maple, we obtained for example h(top0:42;0:33333) 1:74690. Chapter 4 Rotation sets for the resonantly kicked linear oscillator As we have seen in the previous chapters, one can infer a lot about dynamics of a torus map whose rotation set has nonempty interior. We also know that there is an abundance of such maps. (See the example in the section 1.1, Proposition 3.2.3, also [Kwa92].) However, our examples so far were constructed ad hoc to get the rotation set with specic properties. In this chapter we consider a model arising from plasma physics. 4.1 The physical context Consider a particle of mass M and charge e moving in R3 lled with a uniform vertical magnetic eld of induction B0. The force exerted by the eld on the particle is proportional to the vector product of its velocity and the eld. Thus the trajectories are curled into spirals with circular projections on the horizontal plane. More precisely, if x; y; z 2 R are the coordinates of the 87 88 particle (z being the vertical one), then the Newtonian equations of motion are M x = eBc 0 y;_ M y = eBc 0 x;_ M z = 0; where c is the speed of light. The vector of horizontal velocity (x;_ y_ ) rotates with the Larmor frequency ! = eB0 =cM . The three momenta: pz := M z_ , py := M y_ + M!x, and px := M x_ M!y, are conserved. This simple behavior can change radically in the presence of an external electric eld in the system. We will restrict our attention to the case when the electric eld is aligned with the x-axis and does not depend on y; z coordinates (cf. [ZZRSC86b]). These assumptions reduce the number of essential degrees of freedom to one since the momenta pz , py continue to be integrals of the motion. Furthermore, we will require that the eld is a \tight" wave packet with amplitude E (x; t) = P1 TE (x)(t Tn), where T > 0 is xed, E (x) is 2=k-periodic in x, n= 1 and () is the Dirac delta function. The meaning of the formula is that the packet interacts with the particle by boosting its x-velocity by Me TE (x) every T seconds. Thus, if (x_ n; y_n) is the horizontal velocity prior to the kick at the time nT (n 2 Z), then (x_ n+1; y_n+1) = R x_ n + eT E py M y_n ; y_n (4:1) M M! where R is the rotation by the angle = !T . In this way, the analysis of the physical model reduces to investigation of dynamics on R2 generated by the 89 mapping F : (x_ n; y_n) 7! (x_ n+1 ; y_n+1): This reduction was rst carried out in [ZZRSC86b]. What makes the map F interesting is that (even for small amplitudes of E ) numerical experiments and physical heuristic suggest that it exhibits trajectories escaping towards the innity with a nonzero \average acceleration". A signicant physical consequence is an unbounded diusion of plasma particles called also stochastical heating ([ZZRSC86b]): the particles increase their kinetic energy (heat up) by exploiting the energy of the electric eld. In general, there are no rigorous arguments for the existence of the above behavior. A lot of attention is thus devoted to resonant cases when is rational ([ZZRSC86b, CSUZ87, ZZRSC86a]). Among those, the cases of choice are = =2, = =3 or = 2=3 | the only possibilities if we require that some positive iterate of F is a doubly-periodic map of R2, i.e. it commutes with some faithful Z2 -action on R2. Moding out by the Z2 -action yields a torus dieomorphism isotopic to the identity. In terms of the torus map, the presence of stochastical heating corresponds to a nontrivial rotation set. Indeed, in view of (4.1), the rotation set can be interpreted as the totality of asymptotic average accelerations exhibited by the particles (with xed momentum py ). In what follows we will analyze closer the case of = 2=3. (The cases = =3; =2 can be treated analogously.) Our modest goal is to see that, for rather high values of nonlinearity, the rotation set is indeed nontrivial. We calculate it for special cases and provide crude estimate from below in the 90 general case. In the case when is \saw-tooth-like", we prove monotonicity of the rotation set as a function of the amplitude of E (also for rather large amplitudes). The more delicate part of our discussion is greatly simplied by formulation of the problem in terms of a second order recurrence equation. This simple reduction has not been taken up in this context before. We close our discussion by mentioning a toy model in statistical mechanics governed by the recurrence equation. 4.2 The mapping and its rotation set The rotation R2=3 is linearly conjugated to the map L : (u; v) 7! (u; u v); u; v 2 R: By choosing the conjugacy that sends x_ -axis to u-axis and by rescaling the variables, one conjugates F (as prescribed by (4.1)) to G given by G : (u; v) 7! L(u + (v); v) = (v; u v (v)); (4:2) py ) is 2 -periodic. The third iterate of G where (v) = eT=M!k E ( kv + M! commutes with the translations of R2 by vectors in (2Z)2, and so it factors to a torus map g 3 : T2 ! T2 : Even though G and g3 depend on the choice of , we suppress this dependence in the notation since it will be always clear which we are talking about. From now on, if we do not state otherwise, is an arbitrary xed continuous 91 function of period 2. However, we will remember that naturally comes embedded in the two parameter family K; (x) := K(x + ); x 2 R; (4:3) where 2 R and K > 0 correspond to the momentum py and the strength of the forcing respectively. Fact 4.2.1 For any 2 R, replacing () with a shifted function ( + ) + 3 is equivalent to conjugating G by a translation (u; v) 7! (u ; v ). In particular, the rotation set (G3 ) of G3 is unaltered by the shift. For r 2 R, we will denote by Hex(r) the hexagon with vertices (r; 0), L(r; 0) = (0; r), L2 (r; 0) = ( r; r). The following are the most basic properties of the rotation set (G3). Proposition 4.2.1 If G is dened by (4.2), then (i) (G3 ) has a three-fold symmetry, namely (G3 ) = L((G3 )) = L2 ((G3 )); (ii) if is odd, then (G3 ) = (G3 ); (iii) (G3 ) is contained in Hex(sup inf ). Proof of (i). Let Q : R2 ! R2 be given by (u; v) 7! (u + (v); v); so that G = L Q. The displacement of a point p 2 R2 under n + 1 iterates of G is Gn+1(p) p = L Q Gn(p) L Gn(p) + L(Gn(p) p) + Lp p: 92 The rst and the last dierences on the right side are bounded, so (1=(n + 1)) (Gn+1(p) p) is asymptotically equal to L ((1=n) (Gn(p) p)), as n ! 1; and thus (G3 ) = L((G3 )). Q.E.D. Proof of (ii). Just note that if is odd, then Q commutes with central sym- metry p 7! p, and so does the whole map G. Q.E.D. Proof of (iii). Write the displacement of p 2 R2 under G3 as follows G3(p) p = L Q G2(p) G2(p) + L2 Q G(p) G(p) + L3 Q(p) p : The arguments of L; L2; L3 = id, above, belong to a segment I with endpoints (inf ; 0) and (sup ; 0), so G3(p) p sits inside the algebraic (Minkowski) sum L(I ) + L2 (I ) + I , which equals Hex(sup inf ). Q.E.D. Proposition 4.2.1, (ii) can be sharp, as it is shown by the example below. For convenience, we note the following rst. Fact 4.2.2 If (u; v) 2 R2 satises ( v u (v)) = (v) and := (v) (u) 2 2Z, then (u; v) is a lift of a xed point of g3 and (G3 ; (u; v)) = (0; ). Proof. We apply G to (u; v) three times (u; v) 7! (v; u v (v)) 7! ( u v (v); u + (v) ( v u (v))) = ( u v (v); u) 7! (u; v + (v) (u)): Q.E.D. 93 Actually, from the proof, one can see that the inverse of the fact is also true. Example. Consider any normalized so that sup = K and inf = K (see Fact 4.2.1). By Fact 4.2.2, if there are u1; u2; v1 ; v2 such that (v1) = (v2) = ( v1 u1) = K and (u1) = (u2) = ( v2 u2) = K , we conclude that both (0; 2K ) are in the rotation set provided also K = 0 (mod ). By (i) of Proposition 4.2.1, the rotation set is then actually equal to Hex(2K ). It is not dicult to draw a graph of with the required placement of the maxima and the minima. (Use for example u1 = =2, u2 = =6, v1 = =6, v2 = =2.) (One can see that any suitable has to attain one of its global extrema at least three times per period.) Another consequence of Fact 4.2.2 is the following very crude lower bound on the rotation set. Proposition 4.2.2 If l := sup inf 24, then the triangle with vertices Li (0; l=2), i = 1; 2; 3, is contained in the rotation set (G3 ). Proof. Using Fact 4.2.1, we can shift the graph of so that there are v0 ; u0 with (v0) = sup 12 and (u0) = inf 12. Now, as we push v continuously away from v0 (in either direction), before (v) drops below sup =2, the expression 2v (v) changes by at least 4 + 6 = 2, thus sweeping the whole period. In particular, ( 2v (v)) sweeps, in the process, an interval containing [inf ; inf =2]; and so, by the intermediate value theorem, there is v such that l=2 = sup =2 inf =2 (v) ( 2v (v)) 2 2Z. Set u = 2v (v), and see that (u; v) satises the assumption of Fact 4.2.2, 94 thus placing (0; (v) (u)) in the rotation set. By (i) of Proposition 4.2.1, Li (0; (v) (u)), i = 1; 2, are in the rotation set as well. Q.E.D. 4.3 The recurrence, higher period orbits, monotonicity A convenient way of viewing a trajectory of a point (u; v) 2 R2 under G is to label it as a sequence (xk )k2Z so that x0 = u, x1 = v and (xk ; xk+1) = G(xk 1; xk ), for k 2 Z. It is then characterized by the following recurrence equation xk 1 + xk + (xk ) + xk+1 = 0; k 2 Z: (4:4) For n 2 3N, a periodic orbit of G of period n thus corresponds to x = (xk )kn=01 2 Rn satisfying (4.4) with a cyclic indexing modulo n. For = 0, this is when every point is xed by G3 , the two-dimensional space of solutions to (4.4) is conveniently spanned by e = (1; 1; 0; 1; 1; 0; :::; 1; 1; 0); Se = (0; 1; 1; 0; 1; 1; :::; 0; 1; 1); S 2e = ( 1; 0; 1; 1; 0; 1; :::; 1; 0; 1); where S : Rn ! Rn shifts the coordinates cyclically to the right, and S 3e = e. The orbits of G corresponding to periodic orbits of g3 on the torus with nonzero rotation vector come from (xk )k2Z satisfying (4.4) together with the periodicity condition xk+n xk = vk (mod 3) , where v1 ; v2; v3 2 2Z, n 2 3N. To 95 search for them, we may use the following scheme. (Below hx; yi := Pni=1 xiyi, for x; y 2 Rn.) Lemma 4.3.1 For n 2 3N, if z = (zk )kn=01 2 Rn satises zk 1 + zk + (zk ) + zk+1 = k ; 0 k n 1; (4:5) where 2 (2 Z)n and the indexing is cyclic, then there is 2 (2 Z)Z satisfying k 1 + k + k+1 = k (mod n); k 2 Z; (4:6) and the sequence xk := zk (mod n) + k is a solution to (4.4). The displacement over a period is given by xk+n xn = h; S k ei; k 2 Z; (4:7) so the corresponding rotation vector for G3 is (1=n) h; ei; h; Sei . Moreover, one can obtain in this way all of (xk ) 2 RZ corresponding to orbits of G that cover periodic orbits of g3 on the torus. Proof. Solving (4.6) is straightforward: x 0 ; 1 2 2 Z and gure out the rest of i's successively. That xk = zk (mod n) + k satisfy (4.4) follows from putting together (4.5) and (4.6). To calculate the displacement, replace in (4.4) k with k + 1 and subtract unaltered (4.4) to get xk+3 xk = (xk+1) (xk+2) = (zk+1 (mod n)) (zk+2 (mod n)): Sum over the range k; k + 3; :::; k + n to obtain E D xk+n xk = ((zj (mod n)))jk=+kn+1; S k e = h; S k ei; (4:8) 96 D E where we used the fact that (zj 1 (mod n) +zj (mod n)+zj+1 (mod n))jk=+kn+1; S k e = 0. Now suppose that (xk )k2Z covers a periodic orbit of period n 2 3N. Then xk+n xk =: vk (mod 3) 2 2Z, and, because both xk 's and the shifted sequence xk+n's satisfy (4.4), the sequence (vk (mod 3) )k2Z obeys the linear equation vk 1 (mod 3) + vk (mod 3) + vk+1 (mod 3) = 0. This makes k (mod n) := xk 1 (mod n) xk (mod n) (xk (mod n) ) xk+1 (mod n), k 2 Z, well dened. To satisfy (4.5), set zk := xk , k = 0; :::; n 1. Q.E.D. Solving (4.5) for large n is a daunting task regardless of our choice of . We thus have little hope that the rotation set can be calculated exactly for a continuum of values of the parameters K and in (4.3). A way to somehow bridge the particular cases for which the rotation set is nontrivial leads through monotonicity results as the one below. We will call unimodal, if it has only two intervals of monotonicity over its minimal period. We will refer to the endpoints of the maximal intervals of monotonicity as critical points of . Proposition 4.3.1 For a piecewise-smooth unimodal and odd with D := inf x j0(x)j 6, the rotation set of G3 corresponding to K is nondecreasing in K for K 2 [1; +1). Actually, all periodic orbits of the torus map can be continued as K increases. The proof is based on the following \closing lemma". Lemma 4.3.2 Fix > 0 and suppose that z = (zk )kn=01 2 Rn satises zk 1 + zk + K(zk ) + zk+1 = k + k ; 97 where k 2 ( ; ). If, for some open intervals Ik that are free of the critical points of and contain zk , k = 0; :::; n 1, we have d := inf fj1 + K0 (t)j : t 2 Ik ; 0 k n 1g > 2; (4:9) B d d (zk + K(zk )) fz + K(z) : z 2 Ik g; (4:10) and 2 then there is z = (zk ) 2 Rn such that zk 2 Ik , and zk 1 + zk + K(zk ) + zk+1 = k ; 0 k n 1: (4:11) Proof. We obtain z as the limit of a sequence of corrections of z , each di- minishing the supremum of k 's by a denite multiplicative factor < 1. The hypothesis (4:9) guarantees that zk + K(zk ) changes faster (in zk ) than the \coupling term" zk 1 + zk+1, thus k can be eectively decreased by manipulating zk only. The hypothesis (4.10) secures enough \maneuvering space" to perform all innitely many corrections. With this in mind, it will be convenient to think of d d 2 as follows: x 2 (0; 1) and let = 1 + 2=d < 1, then d = = 1 = + + 2 + :::: d 2 2=d 1 (4:12) We describe the rst correction. From (4.12), d d 2 , so, by (4.10), B(zk + K(zk )) fz + K(z) : z 2 Ik g: It follows that we can nd zk0 2 Ik , k = 0; :::; n 1, so that jzk 1 + zk0 + K(zk0 ) + zk+1 + k j = (1 )jk j (4:13) 98 and jzk0 + K(zk0 ) zk K(zk )j = jk j : (4:14) The hypothesis (4.9) implies then that jzk0 zk j d jk j: Putting the two inequalities together we get zk0 1 + zk0 + K(zk0 ) + zk0 +1 = k + 0k ; for some 0k such that j0 j (1 k )jk j + d jk 1j + d jk+1j 1 + 2 d = : In this way n 1 0 0 = max j j : k=0 k (4:15) To repeat the procedure leading from z = (zk ) 2 Rn to z0 = (zk0 ) 2 Rn with z0 as our new z, we need to verify (4.13) with zk and replaced by zk0 and 0 respectively. In view of (4.10) and (4.14), it suces to check that + 0 d d 2 : This is however a consequence of (4.15) and (4.12). It should be clear now how the series in (4.12) unfolds as we need to guarantee the possibility of carrying out the innitely many consecutive steps. Q.E.D. Proof of Proposition 4.3.1. Fix K > 1. We want to prove that the rotation set of G3 corresponding to K contains that corresponding to . Note that, 99 by (i) of Proposition 4.2.1, the rotation set is either equal to f(0; 0)g or it has nonempty interior. In the latter case, by Theorem 0.0.2, the rotation set is the closure of the totality of the rotation vectors of periodic orbits. Consequently, it is enough to show the following: for each periodic orbit of the torus map generated by , there is one with the same rotation vector for K. In view of Lemma 4.3.1, it suces to nd, for z 2 Rn satisfying (4.5), a point z 2 Rn satisfying (4.11). Let M := sup = inf . Clearly, and K have the same intervals of monotonicity, K has larger image, and jK(x) (x)j (K 1)M; for all x 2 R: (4:16) Thus, if z 2 Rn satises (4.5) for some 2 (2Z)n, we can nd for each zk a point yk in the same interval of monotonicity such that K(yk ) = (zk ). By (4.16) and the assumption on the derivative jzk yk j (K 1)M=(KD) (K 1)M=(6K ): In this way yk 1 + yk + K(yk ) + yk+1 = k + k ; with jk j := 3(K 1)M=(6K ) = (K 1)M=(2K ). We will use Lemma 4.3.2 to nd z 2 Rn that obeys (4.11). The hypothesis (4.9) is satised because d D 1 5. Let Ik be the monotonicity interval containing yk , that is Ik = [ck ; c0k ] where ck and c0k are the critical point of closest to yk . To verify hypothesis (4.10), observe that image under K of any 100 maximal interval of monotonicity extends exactly (K 1)M beyond the endpoints of the corresponding image for . This and the fact that jK0j KD yield jK(ck ) + ck K(yk ) yk j jK(ck ) K(yk )j jck yk j 1 1 jK(ck ) K(yk )j 1 KD (K 1)M 1 KD ; and the analogous inequality for c0k . Since d d 2 D 1 (K 1)M D 3 2K , to satisfy (4.10), we need DD 31 (K 2K1)M (K 1)M (1 1=(KD)). Dividing both sides by (K 1)M and setting D = 6, we get a stronger inequality 325K 1 61K , which is true because K 1. Q.E.D. 4.4 A mechanical toy-model Finally, let us point out a mechanical toy-model governed by (4.4). For (zk ) 2 RZ, interpret each zk , k 2 Z, as the position of a unit mass particle along the real axis. Let the particles interact by putting a Hookean string between every two with consecutive indices. Also, subject the particles to a potential force with the potential U (x) = 32 x2 , where is the antiderivative of . In the resulting mechanical model, the net force Fk exerted on the kth particle is the sum of the potential force U 0 (zk ) = 3zk (zk ) and the interaction forces (zk+1 zk ) and (zk 1 zk ) from the two neighbors. Thus we have Fk = zk 1 + zk + zk+1 + (zk ); 101 and the solutions to (4.4) can be interpreted as the stationary congurations of this mechanical system. The construction is somewhat parallel to one relating twist maps on the annulus to the Aubry theory (see e.g. [Ban88, Ang90]). However, for now, we have no interesting theorems that stem from this approach. Appendix A Elementary large deviations for topological dynamics We give a proof of Theorem 2.1.1 that is elementary, it makes no use of ergodic theory. Our original inspiration for this approach stems from Theorem II.6.1 in [Ell85]. However, we present here a point of view not taken up in [Ell85] that yields a very simple argument for Theorem 2.1.1. It can also be used to considerably simplify the proof of Theorem II.6.1 in [Ell85]. Recall that we have f : X ! X , where X is compact and f is continuous. Also, the observable : X ! Rd is a continuous function. Since f and will be xed thruought the rest of the appendix, we will suppress the dependence on them in the most of our notation. In particular, we will write n(x) for Sn(f; )(x), that is n(x) = (x) + ::: + (f n 1(x)). For s 2 Rd and n 2 N, we have the Gibbs weight ascribed to every point x 2 X , namely sn(x) := exphn(x); si. Given a nite set S X , the Gibbs weight sn() determines a measure sn supported on S and given by sn := Px2S sn(x) x, where x is a unit mass at x. 102 103 We are interested in the limiting distribution of the averages of among (; n)-separated orbits counted with respect to the Gibbs weights. (Theorem 2.1.1 pertains to the case s = 0.) The corresponding pressure function ps : X ! R (cf. free energy in [Ell85]) is given by (X ) s s p (t) := lim lim sup(1=n) ln sup exphn(x); tin(x) : S (n; )-separated : !0 n!1 x2S Observe that (t+s) ps(t) = p0(t + s) = ptop (f; ); s; t 2 Rd; (A:1) and so, for the convex conjugate p^s of ps, we have p^s(z) = p^0 (z) hz; si; s; z 2 Rd: (A:2) We should also note that replacing \lim sup" with \lim inf" in the denition of ps yields the same quantity (see (viii) of Theorem 9.4 in [Wal82]). For a set E Rd, we generalize hEtop (f; ) and introduce H+s (E ) := lim lim sup(1=n) ln supfsn(S ) : S (; n)-separated !0 n!1 with n(x)=n 2 E , for all x 2 S g; H s (E ) := lim lim inf(1=n) ln supfsn(S ) : S (; n)-separated !0 n!1 with n(x)=n 2 E , for all x 2 S g: Both H+s (E ) and H s (E ) take values in f 1g [ [0; +1]. Clearly, if E F Rd, then H+s = (E ) H+s = (F ). (Here H+s = stands for either of H+s and H s .) One can also easily verify that, for E; F Rd, H+s (E [ F ) = maxfH+s (E ); H+s (F )g; (A:3) 104 and if H+s (E ) < H s (E [ F ), then H s (F ) = H s (E [ F ), what implies H s (E ) H s (E [ F ) maxfH+s (E ); H s (F )g: (A:4) (v) Mimicking the way htop (f; ) arises from hEtop (f; ), we associate with H+s and H s the following entropies at z 2 Rd: hs+(z) := rlim H s (B (z)); !0 + r hs (z) := rlim H s (Br (z)): !0 (z) Note that h0+(z) = htop (f; ), z 2 Rd. In general, hs+ 6= hs (see the example ahead and Proposition A.0.2). The function z 7! hs+= (z) is quite obviously upper semi-continuous. Also, note that if D := convf(x) : x 2 X g, then (1=n)n(x) 2 D, x 2 X , n 2 N. Consequently, for E Rd, if E \ D = ;, then H+s = (E ) = 1: (A:5) In particular, hs+= (z) = 1; for z 62 D. (Actually, one can take here D := (f; ) | see Proposition 2.1.1.) The dependence of hs+= (z) on s is analogous to that of p^s, namely hs+= (z) = hz; si + h0+= (z); z; s 2 Rd : (A:6) This is because, if r > 0 and S X is a (; n)-separated set with (1=n)n(x) 2 Br (z) for x 2 S , then exp ( nr) jsn(x)= exphnz; sij exp (nr); s 2 Rd; and consequently, exp ( 2nr) j exphnz; si0n(S )=sn(S )j exp (2nr): 105 Lemma A.0.1 For any s 2 Rd, (a) if U Rd is open, then H+s = (U ) supfhs+= (z) : z 2 U g; (b) if V Rd is closed, then H+s (V ) maxfhs+(z) : z 2 V g: Proof. Part (a) is trivial. We prove (b) now. Let D be the convex hull of the image of . From (A.5), we see that H+s (V ) = H+s (V \ D). Thus we can assume that V is compact. Now, set V0 := V , and cover V0 with a nite family A of its compact subsets of diameter less than 1. By (A.3), we have H+s (V0) = maxfH+s (A) : A 2 Ag; denote by V1 an element of A such that H+s (V0) = H+s (V1). Analogously, covering V1 with its compact subsets of diameter less than 1=2, we get V2 V1 with H+s (V1) = H+s (V2). By continuing this procedure, we get a nested sequence of compact sets V0 V1 V2 V3 ::: with diameters diam(Vn) < 1=n and H+s (Vn) = H+s (V ), n 2 N. Thus Tn2N Vn consists of a point, and one can easily verify that, if v is this point, then H+s (v) H+s (V ). Q.E.D. The following is a version of Theorem 2.1.1. Proposition A.0.1 (equivalent of Theorem 2.1.1) If htop (f ) < +1, then (i) convex hull( h0+ ) = p^0, (ii) ( h0+ )^ = p0 . 106 Remark. Suppose that htop (f ) = +1. Then p0 (t) = +1, t 2 Rd, so p^0(z) = 1, z 2 Rd. Also H+s (Rd) = htop (f ) = +1; and, from (b) of Lemma A.0.1, one concludes that convex hull( h0+) is improper (i.e. assumes the value 1). This implies that ( h0+ )^(t) = +1, t 2 Rd | (ii) is satised. In contrast, (i) fails | even though convex hull( h0+ )(z) = 1, for z in the relative interior of the essential domain (cf. Theorem 7.2 [Roc72]); it is +1 or nite outside of this interior. Proof of Proposition A.0.1. First of all parts (i) and (ii) are equivalent: one can derive the other by taking convex conjugates of both sides. In particular, from (ii), after taking convex conjugates, we get (by the involutive property of ^ | see the page 34) that cl(convex hull( h0+ )) = p^0. Since h0+ is lower semi-continuous and has bounded essential domain, we can drop \cl" to get (i) (see Proposition A.0.3). We prove (ii) now. Clearly ps(0) = H+s = (Rd), and so we get from Lemma A.0.1 the following (a) ps(0) hs+(z), s; z 2 Rd, (b) ps(0) supfhs+(z) : z 2 Rdg. Making the dependence on s explicit (via (A.6) and (A.2)), write (a) as p0(s) hz; si + h0+(z). Optimizing over all z 2 Rd, we get p0(s) supfhz; si + h0+(z) : z 2 Rdg = ( h0+ )^(z). Similarly, (b) yields p0 (s) supfhs; zi + h0+ (z) : z 2 Rdg, i.e. p0(s) h^ 0+(s). Hence, h^ 0+(s) = p0(s), what proves (ii). Q.E.D. Simple considerations from convex analysis show that, in Proposition A.0.1, the entropy h0+ can be replaced with h0 . Recall that the set of subdif- 107 ferentials of ps at t 2 Rd is given by @ps(t) := fz 2 Rd : ps(t0 ) ps(t) + hz; t0 ti for all t0 2 Rdg: If @ps (t) consists of a single point, then ps is dierentiable at t and @p(t) = frp(t)g. The following is a convenient characterization (p. 218, [Roc72]): p^s(z) + ps(t) = hz; ti z 2 @ps(t) t 2 @ p^s (z): (A:7) As a corollary, any gradient v := rps(t) is a point of strict convexity of p^s. (Actually, gradients v are exactly the points at which the epigraph epi(p^s) is exposed, i.e. there is a supporting hyperplane to epi(p^s) that touches it only at (v; p^s(v)) | see Corollary 25.1.3, p. 243 in [Roc72].) The following proposition explains the special role of the gradients in our context. Proposition A.0.2 If p0 is dierentiable at s 2 Rd and v := rp0 (s), then h0+(v) = h0 (v) = p^0(v): (A:8) By Th. 18.6 p. 167 in [Roc72], the exposed points are dense among all extremal points. This (using lower semicontinuity of h0 and p^0) leads to the following corollary strengthening Proposition A.0.1. Corollary A.0.1 If htop (f ) < +1, then convex hull( h0 ) = p^0 . Proof of Proposition A.0.2. As we have already said, p^0 is strictly convex at v, and so h0+(v) = convex hull( h0+)(v) = p^0(v): Now, by (A.7), v = rps(0) is the unique point v such that ps(0) = supfhz; 0i p^s(z) : z 2 Rdg = p^s(v): 108 Thus, for any r > 0, maxf p^s+(z) : z 62 Br (v)g < ps(0). However, Lemma A.0.1 yields H+s (Rd n Br (v)) maxfhs+(z) : z 62 Br (v)g; and, since (by Proposition A.0.1) hs+ (z) p^s+ (z), we get H+s (Rd n Br (v)) < ps(0) = H+s = (Rd): By (A.4), H s (Br (v)) = H s (Rd) = ps(0), and so hs (v) = p^s (v). Our claim follows via (A.6) and (A.2). Q.E.D. Remark. For readers familiar with [Ell85], it may be worthwhile to note the following. When p0 (t) is smooth, h0+= (z) = p^0 (z) and Lemma A.0.1 (with s = 0) becomes an analogue of Th. II.6.1 in [Ell85]. Our proof leads along a dierent path than that of Ellis. We start with hs+= , which are almost tautologically the right entropy functions (cf. Def. II.3.1, [Ell85]); and then we show their relation to the free energy ps(t) as a manifestation of the dual behavior of hs+= and ps(t) under variation of s (see (A.6), (A.2)). This approach modies in a straightforward way to Ellis's setting yielding what some may nd to be a simpler and more natural proof of Th. II.6.1. The following example shows that, in general, hs+ 6= hs . Example (cf. the example in the rst section of Chapter 2). For (xi )i2Z 2 f1; 2; 3; 4; 5; 6gZ, (xk ; :::; xl ) is a mod 3 block if xk ; :::; xl are congruent modulo 3. We call it maximal if xk 1; xl+1 are both not congruent modulo 3 to xi , i = k; :::; l. The number l k + 1 is referred to as the length of the block. Fix an arbitrary unbounded subset of natural numbers N . Consider a subshift 109 (; ), where = fx 2 f1; 2; 3; 4; 5; 6gZ : x has at most three maximal parity blocks each either innite or with length in Ng and shifts to the left: (((xi)i2Z))i := xi+1 , i 2 Z. The space is easily seen to be compact and invariant. Let the observable : ! R3 be given by 8 > > > > (1; 0; 0); if x0 = 0 mod 3 > > > > < (x) := > (0; 1; 0); if x0 = 1 mod 3 : > > > > > > > : (0; 0; 1); if x0 = 2 mod 3 For n 2 3N , one has a (1; n)-separated set S with cardinality 2n and such that (1=n)n(x) = (1=3; 1=3; 1=3), x 2 S . Indeed, consider (xi ) 2 that have (x0 ; :::; xn=3 1 ), (xn=3; :::; x2n=3 1 ), (x2n=3 ; :::; xn 1) as mod 3 blocks, where there are 2n=3 possibilities for each block given a residue class mod 3. Thus, h0+(1=3; 1=3; 1=3) ln 2. (Actually, we have equality since one can easily see that htop () = ln 2.) On the other hand, we claim that if N is very sparse, say N := fk!gk2N, then h0 (1=3; 1=3; 1=3) = 1. Indeed, let n = k! 1. For x = (xi ) 2 , any maximal mod 3 block contained in (x0 ; :::; xn 1) has length not exceeding (k 1)!. Thus at least one of the three coordinates of (1=n)n(x) does not exceed (k 1)!=n 1=k. This makes (1=n)n(x) lie in a denite distance from (1=3; 1=3; 1=3) for all x 2 and innitely many n of the form n = k! 1, k 2 N. Our claim follows. 110 We nish with the following fact from convex analysis that we used numerous times. Proposition A.0.3 If f : Rd ! ( 1; +1] is lower semi-continuous and dom(f ) is bounded, then convex hull(f ) is lower semi-continuous. Proof. By lower semi-continuity, f has a nite inmum M 2 R. By considering f M in place of f , we may assume that f is non-negative. Let K := epi(f ) Rd+1. The essential domain dom(f ) is the projection of K on the Rd of the arguments. It is enough to demonstrate that conv(K ) is closed, because then it is the epigraph of convex hull(f ) (see the denitions on p. 36 in [Roc72]). Suppose that (zn; yn) 2 conv(K ) and limn!1(zn; yn) = (z; y). We want to show that (z; y) 2 conv(K ). From the Caratheodory theorem ([Roc72]), +1 = 1 such that there are (wn;i; tn;i) 2 K and n;i; n > 0, Pdi=1 n;i (zn; yn) = dX +1 i=1 n;i (wn;i; tn;i) + (0; n): (A:9) Let B := fi : (tn;i)n2N is unboundedg. Passing perhaps to a subsequence, we can assume that, as n ! 1, we have tn;i ! ti 2 R, i 62 B ; and additionally, zn ! z 2 Rd, n ! 2 R, wn;i ! wi 2 Rd and n;i ! i, i = 1; :::n. Note that there must be i = 0 for i 2 B ; otherwise yn's could not stay bounded. Thus the sum on the right side below (z0 ; y0) := X i62B i (wi; ti) + (0; ); 111 is a convex combination, and z0 = z. Also, y0 y because tn;i 0. Since K is closed, we have (wi; ti) 2 K , so (z0 ; y0) 2 conv(K ). It follows that (z; y) is also contained in conv(K ). Q.E.D. 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Sagdeev, and A.A. Chernikov. Stochastic web and diusion of particles in a magnetic eld. Sov. Phys. JETP, 64:294{303, 1986. Index (; n)-separated set, 29 points of strict convexity, 34 (v) htop (f; ), 30 (v) htop (F ), 69 primitive periodic orbit, 4 proper convex function, 34 relative interior, 34 pseudo-Anosov rel a nite set, 14 ane hull, 34 rotation set, 2 rotation vector, 2 connecting length, 46 scaold of subshifts of nite type, 62 convex hull, 34 critical points, 96 support of v in (; ), 62 displacement, 2 tent function K (v ), 69 eective domain, 34 topological entropy at the asymptotic the set of asymptotic averages, 29 equilibrium state, 38 average, 30 Gibbs weight, 102 topological entropy at the rotation vec- involutive property of convex conju- transfer matrix, 48 tor, 69 gation, 34 transition graph, 45 Minkowski sum, A+B, 55 observable, 29 118