Nonexistence of SBR Measures for some Di eomorphisms
that are \Almost Anosov"
Huyi Hu and Lai-Sang Young*
Department of Mathematics, University of Arizona,
Tucson, AZ 85721
Abstract. The purpose of the paper is to present some simple examples that are
hyperbolic everywhere except at one point, but which do not admit SBR measures. Each
example has a xed point at which the larger eigenvalue is equal to one and the smaller
eigenvalue is less than one.
x0 Introduction
Let f : M ! M be a C
Anosov dieomorphism of a compact connected Riemannian
manifold, and let m denote the Riemannian measure on M . A result of Sinai (see e.g.
S]) says that f admits a unique invariant Borel probability measure with the property
that has absolutely continuous conditional measures on unstable manifolds. This is the
invariant measure that is observed physically, for if : M ! IR is a continuous function,
then for m -a.e. x 2 M ,
Z
;1
1 nX
i
n (f x) ! d
2
i=0
as n ! 1. The dynamical system (f ) is \chaotic" in the following sense: it has positive Lyapunov exponents its metric entropy is equal to the sum of its positive Lyapunov
exponents (f ) is measure-theoretically isomorphic to a Bernoulli shift and it has exponential decay of correlations for Holder continuous test functions. These results have been
extended to Axiom A attractors by Bowen, Ruelle, etc. (See e.g. B].)
In this article we will refer to an invariant measure having absolutely continuous
conditional measures on unstable manifolds as a Sinai-Bowen-Ruelle measure or an SBR
measure. The work of Oseledec, Pesin and others allows us to extend this notion to
a nonuniform setting. (See P] and LS].) While some of the properties of SBR measures
carry over (see e.g. LY], Part I), the question of existence of SBR measures in this broader
context remains poorly understood. We formulate this question more precisely: given a
dieomorphism which appears to be hyperbolic in a large part of phase space, can one
decide whether or not it admits an SBR measure? So far there are very few results outside
of Axiom A, and these results involve delicate estimates. See e.g. BC], BY] for results
on the Henon attractors.
The purpose of the paper is to present some very simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Precise
statements of our results are given in x1. For now imagine slowly deforming a hyperbolic
toral automorphism near the origin O until its derivative has one eigenvalue equal to 1 and
the other eigenvalue less than 1. Our theorem says that for the resulting dieomorphism,
* This research is partially supported by the National Science Foundation.
1
;
P
f i x ! O for almost every x with respect to Lebesgue measure. (x denotes Dirac
measure at x.) This example can be made to be topologically conjugate to the original
toral automorphism, and so it is topologically \chaotic". From the statistical point of view,
however, it is totally deterministic in the sense that for almost every initial condition, the
trajectory spends nearly one hundred percent of its time arbitrarily near the origin O.
Our result can be thought of as a two dimensional version of the following result. Let
f : 0 1] ! 0 1] be a piecewise C 2, piecewise expanding map of the unit interval with
f 0 = 1 at a xed point. Then f cannot admit a nite absolutely continuous invariant
measure. (See PI].) The two dimensional situation is, however, not entirely identical to
that in one dimension, for clearly there exist area preserving dieomorphisms with positive
Lyapunov exponents and nonhyperbolic xed points. A more detailed analysis of whether
or not systems that are \almost Anosov" can admit SBR measures will be given in a
forthcoming paper by the rst named author.
1
n
n 1
i=0
x1 Assumptions and Statements of Results
Let M be a C 1 two dimensional compact manifold without boundary, let m denote
the Riemannian measure on M , and let f 2 Di2(M ). We assume throughout this paper
that f satises the following two conditions.
Assumption I.
1. f has a xed point p, i.e. fp = p.
2. There exist a constant s < 1, a continuous function u with
1
u(x) =
> 1
at x = p ,
elsewhere,
and a decomposition of the tangent space Tx M at every x 2 M into
TxM = Exu Exs
such that
kDf vk kvk
kDf vk (x)kvk
8v 2 E 8v 2 E kDf vk = kvk
8v 2 E :
x
x
and
s
u
s
x
u
x
u
p
p
Assumption II. f is topologically transitive on M .
Denition 1.1. An f ;invariant Borel probability measure on M is called an SBR
measure for f : M ! M if
i) (f ) has positive Lyapunov exponents almost everywhere
ii) has absolutely continuous conditional measures on unstable manifolds.
2
We give the precise meaning of the second condition above.
Let be a measurable partition of a measure space X , and let be a probability
; measure on X . Then there is a family of probability measures fx : x 2 X g with x (x) =
1 such that for every measurable set B X , x ! x(B) is measurable and
B =
Z
X
x(B)d (x):
(1:1)
The family fxg is called a canonical system of conditional measures for and . (For a
reference, see e.g. R].)
Suppose now that f : (M ) ! (M ) has positive Lyapunov exponents almost
everywhere. Then for a.e. x, the unstable manifold W u(x) exists and is an immersed
submanifold of M (see P]). A measurable partition of M is said to be subordinate to
unstable manifolds if for -a.e. x, (x) W u(x) and contains an open neighborhood of x
in W u(x). Let mux denote the Riemannian measure induced on W u(x). We say that has
absolutely continuous conditional measures on unstable manifolds if for every measurable
partition that is subordinate to unstable manifolds, x is absolutely continuous with
respect to mux (written x << mux) for -a.e. x 2 M . (For more details, see e.g. LS]).
It is easy to verify that if x << mux for one measurable partition subordinate
to unstable manifolds, then the same property holds for all other measurable partitions
subordinate to unstable manifolds.
We now state our results. Let f and M be as in the beginning of this section.
Theorem A. f does not admit SBR measures.
Theorem B. For m -a.e. x 2 M ,
n
;1
X
1
lim
f i x = p
n!1 n
i=0
where z is the Dirac measure at z, and the above convergence is in the weak topology.
As a by-product of our proofs for Theorem A and Theorem B, we obtain the following.
Theorem C. f has an innite invariant measure with the following properties:
(i) if U is any open neighborhood of p in M , then (M nU ) < 1
(ii) has absolutely continuous conditional measures on weak unstable manifolds.
Remark 1.2. Weak unstable manifolds are dened in Proposition 2.2 (2). Note that
the denition of absolutely continuous conditional measures on unstable manifolds makes
sense even though is a ;nite measure.
One could think of as an innite SBR measure. In this paper, however, the term
\SBR measure" without any qualications will always be reserved for probability measures.
x2. Preliminaries
Lemma 2.1. The maps x ! fExug and x ! fExs g are continuous.
3
This is an easy consequence of the \gap" between s and 1 = inf fu(x) : x 2 M g.
The proof is left to the reader.
We will use the following notation: for > 0, Exu( ) = fv 2 Exu : jvj g,
Exs ( ) = fv 2 Exs : jvj g, and Ex( ) = Exu( ) Exs ( ).
Proposition
2.2.
There exist two continuous foliations F u and F s on M tangent to
E u and E s respectively for which the following hold.
(1) The leaf of F s through x, denoted by F s(x), is the stable manifold at x, i.e.
F (x) = W (x) = fy 2 M : 9C = C s.t. d(f x f y) C ( ) 8n 0g:
(2) The leaf of F through x, denoted by F (x), is the unstable or \weak unstable"
s
manifold at x, i.e.
s
n
y
u
F
n
s n
u
u
;n ;n
(x) = fy 2 M : nlim
!1 d(f x f y ) = 0g:
(3) There exist constants > 0 and D > 0 such that for all x 2 M , if Fu(x) is
the component of F u(x) \ expx Ex( ) containing x, then exp;x 1 Fu(x) is the graph of a
function ux : Exu( ) ! Exs ( ) with ux(0) = 0 and kuxkC2 D. The analogous statement
holds for Fs(x).
Proof: These results follow from Theorem 5.5 and Theorem 5A.1 in HPS]. We
indicate how F u is obtained.
Let x 2 M be xed. If : Exu( ) ! Exs ( ) is a function with (0) = 0 and Lip() 1,
we let Gx : Efux( ) ! Efsx ( ) be the function dened by
;
graph(Gx) = exp;f x1 f expx(graph )
\E
fx
( ):
Note that Gx is always well dened, even though f is not uniformly hyperbolic. The
function ux in assertion (3) is then obtained as the pointwise limit as n ! 1 of
G ; G ;
f 1x
f n+1 x
G ;
f nx
(0)
where 0 represents the 0 function from Efu;n x( ) to Efs;n x( ), and F u(x) is dened to be
n 0
fn
;
F
u
(f ;n x) :
Remark 2.3.
For convenience we will write W u(x) = F u(x), Wu(x) = Fu(x) etc.
u
u
and refer to W (x) and W (x) as the \unstable manifold" and \local unstable manifold"
respectively at x, even though points on these manifolds may not be contracted exponentially in backwards time.
The Lipschitzness of the W s ;foliation will be very important for us later on. We give
the form of the denition that will be used.
Denition 2.4. Let 1 and 2 be twos W u;leaves, and let : 1 ! 2 sbe a
continuous map dened by sliding along the W ;leaves, i.e. for x 2 1, (x) 2 2 \W (x).
We say W s is Lipschitz if is Lipschitz for every (1 2 ).
4
For y 2 W s(x), let ds(x y) denote the distance between x and y measured along
W s(x), and for z 2 W u(x), let du(x z) be dened similarly.
Proposition 2.5. The W s;foliation is Lipschitz.
In fact, given D1 > 0, there exists
s
L1 > 0 such that for every (1 2 ) with d (x (x)) < D1 8x 2 1, the Lipschitz
constant of is less than or equal to L1.
Proof: This result follows from the stronger statement that the map x ! Exs is C 1,
which can be obtained using the same ideas in the proof of Theorem 6.3 in HP]. We sketch
a more direct proof here for the convenience of the reader.
Let x1 2 1 , and let be an arbitrarily short segment in 1 containing x1 . We will
argue that l( ) l( ), where l denotes length and \" means \up to a constant".
By taking a suitably large iterate of f , we may assume that f n and f n ( ) are
very near each other, and l(f n ) ds(f nx f n (x))
for x 2 . Notice that 8x 2 ,
s n
n
s n
n u
d (f (x) f (x)) D1 ( ) 8n > 0. Also, Dfx jEx 1 8n 0. Observe the following:
(1) l(f n ) l((f n )) n j u 8x 2 (2) Dfxn jExu Dfx
E x
(3) 8y1 y2 2 ,
n n;1
Df j u Y
yn1 Ey1 ;1 const d(f i y1 f i y2 ) ;1 const l(f n )n
Dfy2 jEyu2 i=0
;1 const D1(s )nn const:
(The proof of (3) uses the boundness of the C 2 norms of ux. See Proposition 2.2.3).)
Combining (1)-(3), we get l( ) l( ).
Lemma 2.1 and Proposition 2.2 imply that f has a local product structure, i.e. there
exist constants 0 < < such that 8y z 2 M with d(y z) < , y z] := Wu(y) \ Ws (z)
and z y] := Wu(z) \ Ws(y) each contains exactly one point.
A rectangle R is a set in M such that y z 2 R implies y z] z y] 2 R. If u, s are
segments of W u; and W s ;leaves respectively, then u s] denotes the rectangle fy z] :
y 2 u z 2 sg provided that everything makes sense. If R is a rectangle and x 2 R, we
let W u(x R) = Wu(x) \ R and W s (x R) = Ws (x) \ R. If Q and R are two rectangles,
we say that f n Q u;crosses R if 8x 2 Q with f n x 2 R, f n W u(x Q) \ R = W u(f n x R).
We record a simple fact that will be used in x4.
Proposition 2.6. W u(p) and W s (p) are both dense in M .
Proof: We only prove the proposition for W s(p).
Let P be a rectangle containing p and let R be any other rectangle, both with
nonempty interiors. Let R^ be a strictly smaller rectangle lying in the interior of R. By the
topological transitivity of f , 9n > 0 such that f nR^ \ P 6= . For n suciently large, f n R
is considerably longer than f nR^ in the u;direction. We may therefore assume that f n R
u;crosses P . This implies that f ;n W s(p P ) \ R 6= .
5
x3. Distortion Estimates
The goal of this section is to prove the following.
Proposition 3.1. Given any small rectangle P containing p in its interior, there exist
constants > 0 and J > 1 such that if is a W u;segment with l( ) and \ P = ,
then 8y z 2 and n > 0,
;n u
;
1
J Dfz;n jEzu J:
Dfy jEy
We write yi = f ;i y, zi = f ;i z and i = f ;i . If ; is a W u;segment in P , let
;~ = fp x] : x 2 ;g. In what follows, the letter C will be used to denote a generic
constant, which is allowed to depend only on f .
Lemma+ 3.2. Let P + be one of the components of fP nP , and let = W u(x P + ) for
some x 2 P . Assume that i P for i = 1 n ; 1. Then for any y z 2 ,
;n u
Df j u
z) :
log z;n Ez C d l((y
)
Dfy jEyu
Proof:
First, we have for j n,
;j jY
;1 Df ;1 jE u ; Df ;1 jE u j ;1
X
;1
Df
zi
yi
zi
yi
z jEzu
;1 jE u :
u
1+
C
Df
j
;
Df
log ;j = log
E
zi
yi
zi
yi
Dfy;i 1 jEyu Dfy jEyu
i=0
i=0
i
Using Proposition 2.2 (3) and the Lipschitzness of W s (Proposition 2.5), we see that
;1
Dfzi jEzu
i
; Df ; j Cd (z y ) Cl( ) Cl(~ ):
yi
1
u
Eyui
i
i
i
Since the ~i are pairwise disjoint, we have
j ;1
X
i=1
l(~i ) l(W u(p P )):
The arguments above tell us that 8j n,
du(zj yj ) C du(z y) :
l(j )
l( )
We conclude that
;n n
;1
u
X
Df
z jEzu
log ;n u C du(zj yj ) C d l((z )y) :
Dfy jEy
j =0
6
i
Proof of Proposition 3.1:
Let 0 = n0 = n0 + k0 n1 < n1 + k1 < < nt < nt + kt nt+1 = n be such that
j \ P 6= 8n < j < n + k
i
i
i
1 i t
and
j \ P = otherwise.
Then
;ki
t ni+1;1 ;1 ;n X
t
u Df
j
Dfz jEu
X X
E
u
z
Df
j
n
z
log z;n Ezu = log ;ki i ni +
log ;j1 zj :
Dfy jEy i=1 Dfyni jEyuni i=0 j=ni+ki Dfyj jEyuj
Lemma 3.2 applied to the terms in the rst series gives each a contribution of C 0du(yni zni ),
where C 0 is a constant depending on P . This is summable since du(yni zni ) decreases
exponentially in i. Each term in the second series is less than or equal to Cdu(yj zj ), so
we again have a geometric series.
x4 Proof of Theorem A
The following one-dimensional fact plays a key role.
Lemma 4.1. Let h : ;1 1] ! IR be a C map with h(0) = 0,1h0(0) = 1 and h0(x) 1
8x 2 ;1 1]. Let a 2 0 1], and let a = h; a for i 1. Then P a = 1.
2
0
Proof:
i
i
0
i=0
i
From the Taylor expansion of h, we know that for x > 0,
hx x + Lx2
()
for some L. Increasing L if necessary, we may assume that a1 L1 . We will show that
ai Li1 implies that ai+1 L(i1+1) . Suppose this is not true. Then
ai+1 + La
2
i+1
1
1
1
1
1
1
< L(i + 1) 1 + i + 1 = Li 1 ; i + 1 1 + i + 1 < Li ai contradicting () with x = ai+1.
With ai Li1 8i 1, the desired conclusion follows.
Before giving the proof of Theorem A, we recall some facts from general nonuniform
hyperbolic theory. Let f be an arbitrary C 1+ dieomorphism (not having anything to do
with the situation in this paper), and suppose that f preserves an SBR measure . Let be a partition subordinate to W u. Then it is proved in L] that for -a.e. x, the density
x of x with respect to mx satises
1 ;1 Q
u x(z) = i=0 Dfzi jEzi
1 ;1 x(y) Q
Dfyi jEyu i
i=0
7
for all y z 2 (x). (In particular the quotient on the right makes sense.) The following is
also known to be true. Let f lg be the \Pesin sets", i.e. sets on which f n has uniform
estimates. Then for every l, 9l > 0 such that Wul (x) exists for every x 2 l. Also,
9Cl > 0 such that 8x 2 l,
Cl;1 x((yz)) Cl
8y z 2 W
x
u
l
(x):
(See e.g. P].)
We now return to the situation considered in this paper, i.e. f is again assumed to
satisfy Assumptions I and II.
Proof of Theorem A:
Suppose, to derive a contradiction, that f admits an SBR measure . Then there is a
rectangle R M such that (R \ l) > 0 for some l, and W u(x R) Wul (x) 8x 2 R \ l.
We x a rectangle of the form P = Wu(p) Ws (p)] and let Q = f ;1 P nP . Let be
the partition of Q given by (x) = W u(x Q). An argument similar to that in Proposition
2.6 shows that f n R u;crosses Q for some n > 0. It follows from our discussion above that
there is a set Q^ Q \ f n R with Q^ > 0 such that
(i) x 2 Q^ =) (x) Q^ and
(ii) 9C0 > 0 such that 8x 2 Q^ ,
C0;1 x((yz)) C0
8y z 2 W
x
u
(x Q):
Let
Q(i) = fy 2 Q^ : f j y 2 P for j = 1 ig
and let ~ (i) be the projection of Q(i) onto W u(p P ) by sliding along W s . Then the density
estimate above together with the Lipschitzness of the W s;foliation gives
Q(i) Cl(~ (i)):
Using the facts that f i Q(i), i = 1 2 , are pairwise disjoint subsets of P , and
invariant measure, we have
P
1
X
i=1
(f i Q(i)) =
1
X
i=1
Q(i) C
1
X
i=1
is an
l(~ (i)):
Lemma 4.1 applied to f jWu (p) tells us that this sum diverges, contradicting (M ) = 1.
x5 Proofs of Theorems B and C
We rst construct some neighborhoods of p that are convenient to work with. For a
closed rectangle R, let @ R = fx 2 R : x 2= int W (x R)g.
s
u
8
Lemma
5.1.
There exist rectangles P of the form P = u s] satisfying the follows:
u
s
u
s
(1) and are segments of W (p) and W (p) respectively such that p lies in the
interiors of u and s
segment W^ s W s(p) with f W^ s W^ s such that @ sP W^ s
;(2)^ stheres is a compact
T
and W nW (p P ) int P = (3) there is a compact segment W^ u W u(p) with analogous properties.
Moreover, the diameter of P can be chosen arbitrarily small.
Proof: Use the fact that W u(p) and W s(p) are dense in M . See Lemma 2.6.
We x P as above, and consider the rst return map g : M nP ! M nP . That is, for
x 2 M nP , if (x) is the smallest positive integer with f (x) x 2 M nP , then gx = f (x) x.
Note that g is not dened on a set of Lebesgue measure 0 on M nP , but this will not
concern us.
Lemma 5.2. There exists a g;invariant Borel probability measure with the property that has absolutely continuous conditional measures on the unstable manifolds of
f.
Proof: Let L = W u(x P +), where P + is one of the components of fP nP and
x 2 P +. Let mL be the Lebesgue measure on L and let gnmL be the push-forward of mL ,
i.e., (gnmL)(E ) = mL (g;nE ). We may take to be any accumulation point in the weak nP
;1
topology of n1 gi mL .
i=0
The g;invariance of is clear. To show that it has absolutely continuous conditional
measures, it suces to consider rectangles R in M nP with small diameters. We assume
also that int R \ W^ s = , where W^ s is as in Lemma 5.1. The signicance of the second
condition is as follows: giL is the disjoint union of W u;segments, each one of which begins
and ends at some point in gj W^ s for some j 0. Since f W^ s W^ s, each component of
giL that intersects R u;crosses R. Let i denote the density of gi mL with respect to
Lebesgue measure on giL. Then by Proposition 3.1, 9J > 0 (independent of i) such that
for all x, y in the same component of giL \ R,
J ;1 i ((xy)) J:
i
This bound on densities is passed on to the limit measure .
Proof of Theorem C:
Let Qi = fx 2 M nP : (x) = ig, where is the return time to M nP . Dene
=
1X
i;1
X
i=1 j =0
fj (
j
Qi
):
Then is clearly f ;invariant.
; Tn i Let U be a neighborhood of p. Then M nU is contained in M n
f P for some
i=;n
large n, and this latter set clearly has nite ;measure. This proves that is at most
9
;nite. It cannot be nite because it has absolutely continuous conditional measures on
unstable manifolds, and Theorem A says that f does not admit SBR measures.
Recall that m denotes the Lebesgue measure on M . We now study the asymptotic
behavior of trajectories starting at x for m -a.e. x.
Lemma 5.3. Let g and be as in Lemma 5.2. Then (g ) is ergodic.
Proof: We will follow the two standard steps in the proof of ergodicity of SBR
measures for hyperbolic systems without discontinuities. The rst step is to use Hopf's
argument to show that given a rectangle R, m -a.e. x 2 R is \future-generic" with respect to
some ergodic measure R, with R possibly depending on R. (\Future-genericity" means
nP
;1
R
that n1 gi(x) ! d R as n ! +1 for every continuous function : M ! R.
i=0
\Past-genericity" is dened similarly.) The second step is to show that R = for all R.
Let R M nP be a rectangle. Note that when we use the word \rectangle" or
the symbol \W u(x R)" in this paper, we are always referring to the stable and unstable
manifolds of f | which are not necessarily stable and unstable manifolds of g! First we
need to argue that for suitable R, W u(x R) is indeed a local unstable manifold of g, in the
sense that 8y 2 W u(x R), d(g;nx g;ny) ! 0 as n ! 1. This is true if int R \ W^ s = ,
for this condition will guarantee that 8n 0, f ;n W u(x R) is either entirely in P or it
does not intersect P . Similarly, W s (x R) is a local stable manifold of g if R \ W^ u = .
We recall Hopf's argument (see, e.g. A]) for a rectangle R with the properties in the
last paragraph. Since the conditional measures of are absolutely continuous on unstable
manifolds (Lemma 5.2), there exists L = W u(x R) such that mux -a.e. y 2 L is generic
(both future and past) with respect to some ergodic measure y . All the y 's are in fact
identical because as n ! +1, d(g;ny g;nz) ! 0 8y z 2 L. We call this common
measure R. Now if y is future generic with respect to R, then z is future generic with
respect to R 8z 2 W s(y R). It then follows from the Lipschitzness of the W s ;foliation
(Proposition 2.5) that m -a.e. z 2 R is future generic with respect to R .
To carry out the second step, it suces to observe that if R1 and R2 are rectangles
with the properties above, then 9n > 0 such that gnR1 \ R2 6= .
Proof of Theorem B:
We will show that given arbitrarily small numbers > 0 and > 0, there exist
neighborhoods P2 P1 of p with diam P1 such that for m -a.e. x 2 M nP2 ,
#f0 k n : f n x 2 M nP1 g < #f0 k n : f nx 2 P1nP2g
for all sucient large n.
To see this, let P1 be a rectangle of the type in Lemma 5.1. Let g1 : M nP1 ! M nP1
and 1 be as in Lemma 5.2, and let be the innite measure in the proof of Theorem
C. Let P2 be chosen small enough that (M nP1 ) (P1 nP2). Then 2 := jM nP2 is
invariant under the rst return map g2 : M nP2 ! M nP2 , and (g2 2) is ergodic. The
Birkho Ergodic Theorem applied to (g2 2 ) completes the proof.
10
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