Anosov Flows and Dynamical Zeta Functions
P. Giulietti∗, C. Liverani†and M. Pollicott‡
March 5, 2012
Abstract
We study the Ruelle and Selberg zeta functions for C r Anosov
flows, r > 2, on a compact smooth manifold. We prove several results, the most remarkable being: (a) for C ∞ flows the zeta function
is meromorphic on the entire complex plane; (b) for contact flows
satisfying a bunching condition (e.g. geodesic flows on manifolds of
negative curvature 19 -pinched) the zeta function has a pole at the topological entropy and is analytic and non zero in a strip to its left; (c)
under the same hypothesis as in (b) we obtain sharp results on the
number of periodic orbits. Our arguments are based on the study of
the spectral properties of the transfer operator acting on a suitable
Banach spaces of anisotropic currents.
∗
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy.
e-mail: giuliett@mat.uniroma2.it
†
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy.
e-mail: liverani@mat.uniroma2.it
‡
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.
e-mail: mpollic@maths.warwick.ac.uk
Date: March 5, 2012
2000 Mathematics Subject Classification: 37C30, 37D20, 53D25
Keywords and phrases: Dynamical zeta functions, Anisotropic norms, Transfer operator, Counting of periodic orbits, Dolgopyat Estimates.
We thank V. Baladi, L. Caporaso, M. Carfora, J. de Simoi, S. Gouezel, F. Ledrappier,
A. Sambusetti, R. Schoof, M. Tsujii for helpful discussions and comments. Giulietti and
Liverani acknowledge the support of the ERC Advanced Grant MALADY (246953) and
they also thank the Fields Institute, Toronto, where part of this work was written.
1
1 INTRODUCTION
1
Introduction
In 1956, Selberg introduced a zeta function for a surface of constant curvature
κ = −1 formally defined to be the complex function
ζSelberg (z) =
∞
YY
1 − e−(z+n)λ(γ) , z ∈ C
(1.1)
γ n=0
where λ(γ) denotes the length of a closed geodesic γ. This converges to a
non-zero analytic function on the half-plane Re(z) > 1 and Selberg showed
that ζSelberg has an analytic extension to the entire complex plane, by using
the trace formula which now bears his name [42]. Moreover, he also showed
that the zeroes of ζSelberg correspond to the eigenvalues of the Laplacian. In
fact, the trace formula connects the eigenvalues of −∆ with the information
provided by the geodesics, their lengths and their distribution.1 The definition (1.1) was subsequently adapted to more general settings, including
surfaces of variable curvature, thus giving birth to a new class of zeta functions which we refer to as dynamical zeta functions. However, due to the
lack of a suitable generalized trace formula, few results are known on their
meromorphic extension, the location of their zeroes or their relationships
with appropriate operators.
In 1976, Ruelle [41] proposed generalizing the definition by replacing the
closed geodesics in ζSelberg by the closed orbits of an Anosov flow φt : M → M ,
where M is a C ∞ , d-dimensional Riemannian compact manifold. We recall
that an Anosov flow is a flow such that there exists a Dφt -invariant continuous
splitting T M = E s ⊕ E u ⊕ E c , where E c is the one-dimensional subspace
tangent to the flow, and constants C0 ≥ 1 and λ̄ > 0, such that for all x ∈ M
kDφt (v)k ≤ kvkC0 e−λ̄t
kDφ−t (v)k ≤ kvkC0 e−λ̄t
C0−1 kvk ≤ kDφt (v)k ≤ C0 kvk
if t ≥ 0, v ∈ E s ,
if t ≥ 0, v ∈ E u ,
if t ∈ R, v ∈ E c .
(1.2)
.
.
We will denote by du = dim(Eu ) and ds = dim(Es ). The geodesic flow on
manifolds with negative sectional curvatures are very special examples of
mixing Anosov flows (see [28] and references therein).
1
See [35], and references therein, for a precise, yet friendly, introduction to the Selberg
Trace formula and its relationship with the Selberg zeta function.
2
1 INTRODUCTION
In this context Ruelle defined a zeta function by
Y
−1
ζRuelle (z) =
1 − e−zλ(τ )
,z ∈ C
(1.3)
τ ∈Tp
where Tp denotes the set of prime2 orbits and λ(τ ) denotes the period of the
closed orbit τ . It is known, for weak mixing Anosov flows, that ζRuelle (z)
is analytic and non zero for <(z) ≥ htop (φ1 ) apart for a single pole at z =
htop (φ1 ), where by htop (φ1 ) we mean the topological entropy of the flow (see
[9], or [39, Page 143] for more details). It is easy to see that we can relate
the Ruelle and Selberg zeta functions by
ζRuelle (z) =
ζSelberg (z + 1)
ζSelberg (z)
when they are both defined. Note that it is possible to reconstruct Selberg’s
zeta function from Ruelle’s through the identity
ζSelberg (z) =
∞
Y
ζRuelle (z + i)−1 .
i=0
Here we study ζRuelle . Note that

∞
XX
1 −zmλ(τ ) 
e
= exp 
m
τ ∈Tp m=1
!

ζRuelle (z) =
Y
τ ∈Tp
1 − e−zλ(τ )
−1
(1.4)
X 1
= exp
e−zλ(τ ) ,
µ(τ
)
τ ∈T
where µ(τ ) is the multiplicity of the associated orbit τ and T is the whole
set of periodic orbits on M .
In the very special case of an analytic Anosov flows with real analytic
stable and unstable foliations, Ruelle already showed that his zeta function
has a meromorphic extension to C; and this result was generalized by Fried
2
A periodic orbit τ , of period λ(τ ), is a closed curve parametrized by the flow, i.e.
τ : [0, λ(τ )) → M such that τ (t) = φt (τ (0)). A periodic orbit τ is prime if it is one-to-one
with its image. The range of τ is indicated again by τ . If τp is the prime orbit related to
τ , then µ(τ ) is the unique integer such that λ(τ ) = µ(τ )λ(τp ).
3
CONTENTS
CONTENTS
([17]) still assuming an analytic flow. Here we extend such results to the C ∞
setting.
An additional knowledge on the location of the zeroes of ζRuelle allows one
to gain information on the distribution of the periodic orbits. For example
it is known that if a negatively curved Riemannian manifold has dimension
2 [14] or the sectional curvatures are 41 -pinched [44], then the number N (T )
htop (φ1 )T
of closed orbits of period less than
) + O(ecT )
R x T 1satisfies N (T ) = li(e
where c < htop (φ1 ), and li(x) = 2 ln(s) ds. Note that either the assumption
that M is two dimensional or that the sectional curvatures are 14 -pinched
imply that the horocycle foliation is C 1 . One might then conjecture that
such a foliation regularity is necessary in order to obtain a sharp error term.
However, we show that this is not the case and, although we cannot prove it
in full generality, we conjecture that the above bound holds for all contact
Anosov flows.
The structure of the paper is as follows. In section 2 we present the
statements of the main results in this paper. We also explain the strategy of
the proofs assuming several Lemmata and constructions explained detail in
later sections. In section 3, we construct the spaces on which our operators
will act. Sections 4, 5 and 6 contain estimates for transfer operators and their
“flat trace”. In sections 7 and 8 we restrict ourself to the case of contact flows.
In section 7 we exclude the existence of zeroes in a vertical strip to the left
of htop (φ1 ). In sections 8 we obtain a bound on the growth of ζRuelle in this
strip.
In Appendix A we collect together, for the reader’s convenience, several
facts from differential geometry, while in Appendix B we discuss the orientability of the stable distribution. In Appendix C we relate the topological
entropy and the volume growth of manifolds. Finally, in Appendix D, we
recall some necessary facts concerning holonomies.
Contents
1 Introduction
2
2 Statement of Results
2.1 Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Theorems and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
3 Cones and Banach spaces
15
3.1 Charts and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4
2 STATEMENT OF RESULTS
3.2
3.3
Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Averaging operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Transfer operators
25
4.1 Properties of the transfer operator . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Properties of the Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Proof of the Lasota-Yorke . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Flat Traces
38
6 Tensorial Operators
43
6.1 Spaces and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.2 Lasota-Yorke inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.3 Trace representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7 Contact flows
57
7.1 Dolgopyat’s estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.2 A key (but technical) inequality . . . . . . . . . . . . . . . . . . . . . . . . . 65
8 Growth of ζ-functions
69
Appendices
70
A External Forms: a toolbox
70
B Orientability
72
C Topological Entropy and volume growth
76
D Holonomies
79
References
81
2
2.1
Statement of Results
Some notation
We use Bd (x, r) to designate the open d-dimensional ball with center x and
radius r.
We will use C# to represent a generic constant (depending on the manifold, the flow and the choice of the charts and the partition of unity made in
Section 3) which could change from one occurrence to the next, even within
the same equation. We will write Ca1 ,...,ak for generic constants depending
5
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
on the parameters a1 , . . . , ak , which could still change from time to time.
Finally, numbered constants C0 , C1 , C2 , . . . are constants with fixed value.
2.2
Theorems and Proofs
Our first result applies to all Anosov flows φt on a connected, compact and
orientable Riemannian manifold M .
Theorem 2.1. For any C r Anosov flow φt with r > 2, ζRuelle (z) is meromorphic in the region
λ r−1
<(z) > htop (φ1 ) −
2
2
where λ, is determined by the Anosov splitting, and bxc denotes the integer
part of x. Moreover, ζRuelle (z) is analytic for <(z) > htop (φ1 ) and non zero
for <(z) > max{htop (φ1 ) − λ2 r−1
, htop (φ1 ) − λ}. If the flow is topologically
2
mixing then ζRuelle (z) has no poles on the line {htop (φ1 ) + ib}b∈R apart from
a single simple pole at z = htop (φ1 ).
Corollary 2.2. For any C ∞ Anosov flow the zeta function ζRuelle (z) is meromorphic in the entire complex plane.
Note that if the flow is not weak mixing then the flow can be reduced
to a constant ceiling suspension and hence there exists b > 0 such that
ζRuelle (z + ib) = ζRuelle (z) (for more details see [38]).
Corollary 2.3. ζRuelle (z) and ζSelberg (z) are meromorphic in the entire complex plane for smooth geodesic flows φt on any connected compact orientable
Riemannian manifold with variable strictly negative sectional curvatures.3
Moreover, the zeta functions ζRuelle (z) and ζSelberg (z) have no zeroes or poles
on the line {htop (φ1 ) + ib}b∈R , except at z = htop (φ1 ) where both ζRuelle (z)−1
and ζSelberg (z) have a simple zero.
3
This provides an answer to an old question of Smale: “’Does ζSelberg (z) have nice
properties for any general class of flows?” cf. pages 802-803, of [43]. Smale also specifically asked if for suspension flows over Anosov diffeomorphisms close to constant height
suspensions the zeta function ζSelberg (z) has a meromorphic extension to all of C. The
above theorem answers these questions in the affirmative for C ∞ Anosov flows, despite
Smale’s comment “I must admit that a positive answer would be a little shocking”.
6
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
Next, we specialize to contact Anosov flows. Let λ+ ≥ 0 such that
kDφ−t k∞ ≤ C0 eλ+ t for all t ≥ 0.
Theorem 2.4. For any C r , r > 2, contact flow with λλ+ ≥ 31 there exists
τ∗ > 0 such that the Ruelle zeta function is analytic in {z ∈ C : <(z) ≥
htop (φ1 ) − τ∗ } apart from a simple pole at z = htop (φ1 ).
Remark 2.5. Note that the bunching condition λ2λ+ ≥ 23 implies that the
invariant foliations are at least 32 -Hölder continuous. Also remember that an
2
√
a-pinched geodesic flow is (2 a + ε)-bunched, that is a ≤ λλ+ + ε, see [22]
for more details.
The above fact has several important implications. To begin with some
low-hanging fruits remember that the determinants for 0-forms and d-forms
have their first zeros at <(z) = 0 (since they are exactly the dynamical
determinants of the usual Ruelle transfer operator). Thus
Corollary 2.6. For a volume preserving three dimensional Anosov flow we
have that the zeta function ζRuelle (z) is meromorphic in C and, moreover,
• ζRuelle (z) is analytic for <(z) ≥ htop (φ1 ) − τ∗ , except for a pole at
htop (φ1 );
• ζRuelle (z) is non-zero for <(z) > 0.
Moreover, Theorem 2.4 allows us to extend results of Huber-Selberg (for
constant sectional curvatures), Pollicott-Sharp (for surfaces of negative curvature) and Stoyanov (for 41 -pinched geodesic flows).
Theorem 2.7 (Prime Geodesic Theorem with exponential error). Let M be
a manifold better than 19 -pinched, with strictly negative sectional curvature.
Let π(T ) denote the number of closed geodesics on M with length at most T ,
then there exists δ > 0 such that
π(T ) = li(ehtop (φ1 )T ) + O(e(htop (φ1 )−δ)T ) as T → +∞
Remark 2.8. The above Theorems are most likely not optimal. The 91 pinching can conceivably be improved with some extra work (one would need
to improve, or circumvent the use of, Lemma 7.7) but we do not see how to
remove such conditions completely, even though we believe it to be possible.
7
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
Let us start the discussion of the proofs. The basic objects we will study
are the dynamical determinants, following the approach introduced by Ruelle
[41], which arise naturally in the dynamical context and are formally of the
general form
!
X tr ∧` (Dhyp φ−λ(τ ) ) e−zλ(τ )
,
(2.1)
D` (z) = exp −
µ(τ )(τ ) det 1 − Dhyp φ−λ(τ ) τ ∈T
where (τ ) is 1 if the flow preserves the orientation of Es along τ and −1
otherwise. The symbol Dhyp φ−λ(τ ) indicates the derivative, on the orbit, of
the return map to a local transverse section to the orbit τ . By ∧` A we mean
the matrix associated
to the standard `-th
exterior product of A. Note that
`
tr ∧ (Dhyp φ−λ(τ ) ) , det 1 − Dhyp φ−λ(τ ) and det(Dφ−λ(τ ) E s ) depend only
on the orbits (see comments before
equation 5.9 for more details). Also, note
that (τ ) = sign det(Dφ−λ(τ ) E s ) . The sum in (2.1) is well defined provided
<(z) is large enough.
As a direct consequence of the linear algebra identity for a n × n matrix4
A
n
X
(−1)` tr(∧` A)
(2.2)
det(1 − A) =
`=0
we obtain from (1.4), (2.1) and (2.2) a product formula à la Atiyah-Bott.
Thus, we deduce that5
−zλ(τ ) !
d−1
d−1 X
`
`+ds
Y
X
tr
∧
(D
φ
)
e
(−1)
`+d
+1
hyp
−λ(τ
)
s
D` (z)(−1)
= exp
det 1 − Dhyp φ−λ(τ ) µ(τ
)(τ
)
`=0
`=0 τ ∈T
!
d
s
X (−1) det 1 − Dhyp φ−λ(τ ) e−zλ(τ )
= exp
(2.3)
det 1 − Dhyp φ−λ(τ ) µ(τ
)(τ
)
τ ∈T
!
X e−zλ(τ )
= exp
= ζRuelle (z).
µ(τ )
τ ∈T
Thus Theorem 2.1 follows by the analogous statement for the dynamical
determinants D` (z). To study the region in which the D` (z) are meromorphic
we will proceed in the following roundabout manner. First we define the
following objects.
4
5
See, for example, [47, Section 3.9]
for more details.
Indeed, sign det 1 − Dhyp φ−λ(τ ) = (−1)ds sign det( Dφ−λ(τ ) E s ) = (−1)ds (τ ).
8
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
Definition 2.9. Given 0 ≤ ` ≤ d − 1, τ ∈ T let
.
χ` (τ ) =
tr ∧` (Dhyp φ−λ(τ ) )
.
(τ ) det 1 − Dhyp φ−λ(τ ) (2.4)
Moreover, for ξ, z ∈ C, we let
∞
X
ξ n X χ` (τ )
.
e
λ(τ )n e−zλ(τ )
D` (ξ, z) = exp −
n!
µ(τ
)
n=1
τ ∈T
!
.
(2.5)
Note that the series in (2.5) trivially converges for |ξ| sufficiently small and
<(z) sufficiently large.
Lemma 2.10. Let 0 ≤ ` ≤ d − 1, ξ, z ∈ C, <(z) sufficiently large and |ξ − z|
sufficiently small. Then we can write
e ` (ξ − z, ξ) = D` (z) .
D
D` (ξ)
(2.6)
Proof. The proof is by a direct calculation
!
∞
n X
X
(ξ
−
z)
χ
(τ
)
`
n
−ξλ(τ
)
e ` (ξ − z, ξ) = exp −
D
λ(τ ) e
n!
µ(τ )
n=1
τ ∈T
!
X χ` (τ )
D` (z)
= exp −
e−zλ(τ ) − e−ξλ(τ )
.
=
µ(τ )
D` (ξ)
τ ∈T
Hence Theorem 2.1 is implied by the following.
Proposition 2.11. For any C r Anosov flow, with r > 2, there exists A >
0 such that, for each <(ξ) ∈ [A, 2A], D` (ξ) is analytic and the function
e ` (ξ − z, ξ) is meromorphic for z in the region
D
λ r−1
|ξ − z| < <(z) − htop (φ1 ) + |ds − `|λ +
.
2
2
9
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
We can then freely move ξ along the line {a + ib}b∈R and we obtain that
e
.
D` (ξ − z, ξ) is meromorphic for <(z) > htop (φ1 ) − |ds − `|λ − λ2 r−1
2
Ruelle’s original proof of his result on zeta functions was based on Markov
partitions and Grothendieck’s theory of nuclear operators, and hence required analyticity. Here instead, to deal with the finite smoothness case, we
prove Proposition 2.11 using the method of extending determinants through
the choice of anisotropic Banach spaces, based on the approach of LiveraniGouezel [19], Butterley-Liverani [10] and Liverani-Tsujii [32]. This approach
allows us to apply transfer operator methods directly to the manifold M by
resorting to currents, avoiding the complications, due to the lack of regularity
of the foliations, present in previous approaches.
Remark 2.12. Recently alternative approaches have been developed based
on different Banach spaces or on different strategies to compute dynamical
determinants for Anosov diffeomorphisms. It is possible that such methods
could be adapted to the present case and that such versions might allow one to
obtain sharper estimates when one has finite differentiability (as in BaladiTsujii [6]) or when one can exploit an additional contact structure (see Tsujii
[46]).
Proof of Proposition 2.11. Let Ω`s (M ) be the space of `-forms on M , i.e.
the C s sections of Λ` (T ∗ M ). Let Ω`0,s (M ) ⊂ Ω`s (M ) be the subspace of forms
null in the flow direction (see (3.5)). In section 3 we construct a family of
Banach spaces B p,q,` as the closure of Ω`0,s (M ) with respect to a suitable
anisotropic norm so that the spaces B p,q,` are an extension of the spaces in
[19].6 Such spaces are canonically embedded in the space of currents (see
Lemma 3.4).
In section 4 we define a family of operators for h ∈ Ω`0,s (M ), t ∈ R+ as7
.
(`)
Lt (h) = φ∗−t h.
(2.7)
Remark 2.13. In order to simplify a rather involved argument we chose to
give full details only for the case in which the invariant foliations are orientable (i.e., (τ ) = 1 for all τ ). Notably, this includes some of the most
6
The indexes p, q measure, respectively, the regularity in the unstable and stable
direction.
(`)
7
Note that by restricting the transfer operator Lt to h ∈ Ω`0,s (M ) we mimic the action
of the standard transfer operators on sections transverse to the flow, in fact we morally
project our forms on a Poincaré section. Moreover we adopt the standard notation where
f ∗ denotes the pullback and f∗ indicates the push-forward.
10
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
interesting examples, such as geodesic flows on manifolds with strictly negative sectional curvature (see Lemma B.1). To treat the non orientable case
it is often sufficient to slightly modify the definition of the operator (2.7)
by introducing an appropriate weight, see equation (B.1), and then repeating
almost verbatim the following arguments. Unfortunately, as far as we can
see, to treat the fully general case one has to consider more general Banach
spaces than the ones used here. This changes very little in the arguments
but makes the notation much more cumbersome. The reader can find the
essential details in Appendix B.
The operators (2.7) generalize the action of the transfer operator Lt on
(`)
the spaces B p,q of [19]. We prove in Lemma (4.3) that the operators Lt
satisfy a Lasota-Yorke type estimate for sufficiently large times. To take care
of short times we restrict ourself to a new space Bep,q,` which is the closure
of Ω`0,s (M ) with respect to a slightly stronger norm |||·|||p,q,` , in this we follow
[5].
(`)
On Bep,q,` the operators Lt form a strongly continuous semigroup with
.
generators X (`) (see Lemma 4.5). We can then consider the resolvent R(`) (z) =
(z1 − X (`) )−1 . The cornerstone of our analysis is that, although the operator
X (`) is an unbounded closed operator on Bep,q,` , we can access its spectrum
thanks to the fact that its resolvent R(`) (z) is a quasi compact operator on
the same space. More precisely, in Lemma 4.7 we show that for ξ ∈ C,
<(z) > htop (φ1 ) − |ds − `|λ̄, p + q < r − 1 the operator R(`) (z) has essential
spectral radius
−1
ρess (R(`) (z)) ≤ <(z) − htop (φ1 ) + |ds − `|λ + λ min{p, q} .
We can then write R(`) (z) = P (`) (z) + U (`) (z) where P (`) (z) is a finite rank
operator and U (`) (z) has spectral radius arbitrarily close to ρess (R(`) (z)). In
section 5, we define a “flat trace”8 denoted by tr[ . In Lemma 5.1
we show
that for <(z) large enough and n ∈ N, we have that tr[ R(`) (z)n < ∞ and
tr[ R(`) (z)n =
X χ` (τ )
1
λ(τ )n e−zλ(τ ) .
(n − 1)! τ ∈T µ(τ )
(2.8)
8
Such terminology has also been adopted by the dynamical system community, following the work of Atiyah and Bott ([3],[4]), since, as the reader will see, it is morally a
regularization, or a flattening, of the trace.
11
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
Furthermore, in Lemma 6.10, we prove that, for each λ < λ and n ∈ N,9
[ (`) n
(`)
n
tr (R (z) ) − tr P (z) ≤Cz,λ <(z) − htop (φ1 ) + |ds − `|λ
(2.9)
−n
λ
.
+ min{p, q}
2
where “tr” is the standard trace.
Remark 2.14. Given the two previous formulae one can have a useful heuristic explanation of the machinery we are using. Indeed, (2.9) shows that
tr[ (R(`) (z)) is essentially a real trace, then substituting formula (2.8) in (2.1)
and performing obvious formal manipulations we have that D` (z)−1 can be
interpreted as the “determinant” of R(`) (z). Moreover, to connect our objects
to the more traditional ones, note that the action on ds forms corresponds
to the usual transfer operator with a weight given by the unstable determinant. Hence, the maximal eigenvalue of X (ds ) is htop (φ1 ) and the associated
eigenvector is given by the measure of maximal entropy. On the contrary,
X (0) and X (d) are associated to the usual transfer operator and its adjoint.
Thus, their maximal eigenvalue is zero and the associated eigenvector yields
the SRB measure. Also, as usual, the other eigenvalues are connected to the
poles of the Laplace transform of the correlations.
Note that if ν ∈ σ(P (`) (z)) \ {0}, then z − ν −1 ∈ σ(X (`) ).10 Thus, ν =
(z−µ)−1 where µ ∈ σ(X (`) ). Let ξ ∈ C such that a = <(ξ) is sufficiently large
so that D` (ξ) is well defined. Let ρp,q,` < a−htop (φ1 )+|ds −`|λ+ λ2 min{p, q}.
9
The loss of a factor 2 in the formula (2.9) is due to an artifact of the method of proof.
We live with it since it does not change substantially the result and to obtain a sharper
result might entail considerably more work.
10
As usual, we denote by σ the spectrum.
12
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
Let λi,` be the eigenvalues of X (`) . For each z ∈ B2 (ξ, ρp,q,` ),
!
∞
n
X
(ξ
−
z)
[
(`)
n
e ` (ξ − z, ξ) = exp −
tr (R (ξ) ) =
D
n
n=1



∞
n
X
X
(ξ
−
z)
1


= exp −
+ O Cξ,λ ρ−n
p,q,`
n
n
(ξ
−
λ
)
i,`
n=1
λi ∈B2 (ξ,ρp,q,` )


X
∞
n
X
(ξ − z)
ξ−z

+
O Cξ,λ ρ−n
= exp −
log 1 −
p,q,`
ξ − λi
n
n=1
λi ∈B2 (ξ,ρp,q,` )


Y
ξ − λi 
=
ψ(ξ, z)
z − λi
λi ∈B2 (ξ,ρp,q,` )
(2.10)
where ψ(ξ, z) is analytic and non zero for z ∈ B2 (ξ, ρp,q,` ). The results follows
by optimizing the choice of p, q.
Once Theorem 2.1 is established we can use the above machinery to obtain
more information on the location of the zeros.
Proof of Theorem 2.4. In Lemma 7.10 we prove that there exists γ, C1 >
0 such that, for q < λλ+ , <(z) > htop (φ1 ) and n ≥ C1 ln |=(z)|,
(d ) n R s (z) 1,q,ds
≤ C# (<(z) − σds )−n |=(z)|γ .
By the resolvent identity
R
(ds )
(z − a) =
∞
X
an R(ds ) (z)n ,
n=0
which implies the result.
Proof of Theorem 2.7. The proof is then based on the following estimate,
established in Lemma 8.2. There exists γ1 ∈ (0, 1), B > 0 such that, for each
z ∈ C, htop (φ1 ) ≥ <(z) > htop (φ1 ) − τ2∗ , |=(z)| ≥ B, we have that
d
ln ζRuelle (z) ≤ C# |z|γ1 .
(2.11)
dz
13
2.2 Theorems and Proofs
2 STATEMENT OF RESULTS
Starting from (2.11) we follow a classical approach in number theory. Let
X
.
ψ(T ) =
htop (φ1 )λ(τ )
enhtop (φ1 )λ(τ ) ≤T
.
ψ1 (T ) =
Z
T
X
ψ(x)dx =
1
.
π0 (T ) =
enhtop (φ1 )λ(τ ) ≤T
.
1 and π1 (T ) =
X
nhtop (φ1 )λ(τ )
e
htop (φ1 )λ(τ )(T − enhtop (φ1 )λ(τ ) )
≤T
X
e
htop (φ1 )λ(τ )
1.
≤T
First recall the following simple complex integral. For d > 1 we have that
(
Z d+i∞
0
if 0 < y ≤ 1
y z+1
1
(2.12)
dz =
2πi d−i∞ z(z + 1)
y − 1 if y > 1
If we denote ζ0 (z) = ζRuelle (htop (φ1 )z) then we can write
Z
d+i∞
ψ1 (T ) =
d−i∞
ζ 0 (z)
− 0
ζ0 (z)
T z+1
dz
z(z + 1)
(2.13)
for any d > 1 sufficiently large. This comes by a term by term application of
(2.12) to
∞
ζ00 (z) X X
=
htop (φ1 )e−znhtop (φ1 )λ(τ ) .
−
ζ0 (z) n=1 τ
Moving the line of integration from Re(z) = d > 1 to Re(z) = c < 1 gives
Z c+i∞ 0 T2
ζ (z)
T z+1
ψ1 (T ) =
+
− 0
ds
(2.14)
2
ζ0 (z) z(z + 1)
c−i∞
for c = htop (φ1 ) − δ < 1 sufficiently close to 1 so that we can apply the
2
Dolgopyat type estimate (2.11). Note that the term T2 comes from the line
of integration crossing the pole for ζ(z) (now at z = 1). Moreover, the
integral yields a factor T c+1 i.e. ψ1 (T ) = 21 T 2 + O(T c+1 ). Next we have the
following asymptotic estimates.
Lemma 2.15. We can estimate, provided 1 > d > c > 0, that
ψ(T ) = T + O(T (c+1)/2 );
14
3 CONES AND BANACH SPACES
π0 (T ) = li(T ) + O
T (c+1)/2
log T
;
π1 (T ) = li(T ) + O(T (d+1)/2 ).
Since the proof of the above Lemma is completely analogous to the case
of prime numbers, we omit it and refer the reader to [15, Chapter 4, Section
4]. Finally, we can write
π(T ) = π1 ehtop (φ1 )T = li(ehtop (φ1 )T ) + O(e[htop (φ1 )(d+1)/2]T )
and the theorem follows with htop (φ1 )(d + 1)/2 = htop (φ1 ) − δ.
3
Cones and Banach spaces
We want to introduce appropriate Banach spaces of currents11 over a ddimensional smooth12 compact Riemannian manifold M .13 The basic idea,
going back to [33], is to consider regular “objects” that are “smooth” in
the unstable direction while being “distributions” in the stable direction.
This is obtained by defining norms in which such objects are integrated,
against smooth functions, along manifolds close to the stable direction. Unfortunately, the realization of this program is fairly technical, since we have
to define first the class of manifolds on which to integrate, then determine
which are the relevant objects and, finally, explain what we mean by regular.
3.1
Charts and Notation
We start with some assumptions and notation. More precisely, for some
r ∈ N, there exists δ0 > 0, such that, for each δ ∈ (0, δ0 ) and ρ ∈ (0, 4), there
11
See Federer [16, Sections 4.1.1 - 4.1.7] for a detailed presentation of currents.
One could easily generalize to C r manifolds at the price of keeping track of the
regularity of the manifold throughout the computations. We avoid doing so for readability.
13
It is very convenient to outline a general construction of such spaces based only on
an abstract cone structure, since later on we will apply this construction twice: once on
M for certain operators, and once on M 2 for some other operators.
12
15
3.1 Charts and Notation
3 CONES AND BANACH SPACES
exists an atlas {(Uα , Θα )}α∈A , where A is a finite set, such that14
p

Θα (Uα ) = Bd (0, 30 δ 1 + ρ2 )



∪α Θ−1
α (Bd (0, 2δ)) = M
−1
Θα ◦ Θ−1

β (x̃, xd + s) = Θα ◦ Θβ (x̃, xd ) + (0, s)


kΘα ◦ Θ−1
β kC r ≤ 2.
(3.1)
Note that the above implies15 the existence of a global vector field V such
that Θα are flow box charts for the flow generated by V ; i.e. for all α ∈ A,
∂
(Θ−1
α )∗ ( ∂xd ) = V.
Let {ψα }α∈A be a smooth partition of unity subordinate to our atlas such
that supp(ψα ) ⊆ Θ−1
= 1. In addition,
α (Bd (0, 2δ)) ⊂ Uα and ψα |Θ−1
α (Bd (0,δ))
we assume d1 + d2 = dim(M ) are given, and let
.
Cρ = {(s, u) ∈ Rd1 × Rd2 : kuk < ρksk}.
(3.2)
Next, we assume that there exists 4 > ρ > ρ− > 0 such that, for all α, β ∈ A,
C0 ⊂ (Θα )∗ ◦ (Θ−1
β )∗ Cρ− ⊂ Cρ ,
(3.3)
when it is well defined. Note that, by compactness, there must exists ρ1 > 0
such that Cρ1 ⊂ (Θα )∗ ◦ (Θ−1
β )∗ Cρ− .
This concludes the hypotheses on the charts and the cones. For each
` ∈ {0, . . . , d}, let Λ` (T ∗ M ) be the algebra of the exterior `-forms
Pon M . We
.
`
r
`
∗
write
Ω
(M
)
for
the
space
of
C
sections
of
Λ
(T
M
).
Let
h
=
r
α∈A ψα h =
P
`
α∈A hα so that hα ∈ Ωr (Uα ).
Let {e1 , . . . , ed } be the canonical basisn of Rd . For all α ∈ A and
o x ∈
∂
∂
−1
−1
Uα consider the basis of Tx Uα given by (Θα )∗ ∂x1 , . . . , (Θα )∗ ∂xd . Let
{êα,1 , . . . , êα,d } be the orthonormal basis of Tx Uα obtained from the first
one by applying the Gram-Schmdit
procedure,
setting as first element of the
−1 ∂ ∂
−1
algorithm êα,d = (Θα )∗ ∂xd / (Θα )∗ ∂xd . Let ωα,1 , . . . , ωα,d be the dual basis
of 1-forms such that ωα,i (êα,j ) = δi,j . Thus we can define a scalar product on
14
The relations are meant to be valid where the composition is defined. The k · kC r
norm is precisely defined in (3.6). Note that the explicit numbers used (e.g. 2, 30, . . . ) are
largely arbitrary provided they satisfy simple relations that are implicit in the following
constructions.
15
To see that the first, second and third relation can always be satisfied consider charts
determined by the exponential map.
16
3.2 Banach Spaces
3 CONES AND BANACH SPACES
Tx∗ Uα by the expression hωα,i , ωα,j i = δi,j . Note that the above construction
“respects” the special direction V .
Let {ī = (i1 , . . . , i` ) ∈ {1, . . . , d}` : i1 < i2 < . . . < i` } be the set
.
I` of `-multiindices ordered by the standard lexicographic order. Let eī =
.
ei1 ∧ · · · ∧ ei` in Λ` (Rd ) and dxī = dxi1 ∧ · · · ∧ dxi` ∈ (Λ` )(Rd ) so that
dxī (ej̄ ) = δī,j̄ . Let {eα,ī } ⊂ Λ` (Tx Uα ) and {ωα,ī } ⊂ Λ` (Tx∗ Uα ) be defined in
the same way.
∗
`
Given h ∈ Ω`r (Uα ) and (Θ−1
α ) h ∈ Ωr (Bd (0, 12δ)) we will write
X
h=
hα,ī ωα,ī .
ī∈I`
We define the scalar product
hh, giΩ`
.
=
Z
hh, gix ω(x)
(3.4)
M
where ω is the Riemannian volume form and hh, gix is the usual scalar product for forms16 (see (A.1) for a precise definition).
In the sequel we restrict ourselves to
. Ω`0,r (M ) = h ∈ Ω`r (M ) : h(V, . . .) = 0 .
(3.5)
¯
For f : Bd (0, δ) → Rd , d¯ ∈ N, we use the following C r -norm17


if r = 0
supx kf (x)k
kf (x)−f (y)k
kf kC r = kf kC 0 + supx,y∈Bd (0,δ) kx−ykr
if 0 < r < 1

Pbrc−1 brc−i
kDi f kC 0 + Dbrc f C r−brc if r ≥ 1
i=0 2
3.2
(3.6)
Banach Spaces
Given the above setting, we are going to construct several Banach spaces.
The strategy is to first define appropriate norms and then close the space
16
In the following we will drop the subscript x in the scalar product whenever it does
not create any confusion.
17
By kDk f (x)k we mean the sup of all the possible derivative of order k of f at x.
Remember that P
for such a C r norm, for r ∈ R+ , we have kf gkC r ≤ kf kC r kgkC r and
r
kf ◦ gkC r ≤ C# i=0 kf kC r kDgkC r−1 · · · kDgkC r−i .
17
3.2 Banach Spaces
3 CONES AND BANACH SPACES
of `-forms in the associated topology. For sufficiently large L > 0, let us
define18
.
Fr (ρ, L) = {F : Bd1 (0, 6δ) → Rd2 : F (0) = 0;
(3.7)
kDF kC 0 (Bd1 (0,6δ)) ≤ ρ ; kF kC r (Bd1 (0,6δ)) ≤ L}.
For each F ∈ Fr (ρ, L), x ∈ Rd , ξ ∈ Rd1 , let Gx,F (ξ) : Bd1 (0, 6δ) → Rd be
.
defined by Gx,F (ξ) = x + (ξ, F (ξ)).
.
e L) =
Let us also define Σ(ρ,
{Gx,F : x ∈ Bd1 (0, 2δ), F ∈ Fr (ρ, L)}.19 For
e we define the leaf20 Wα,G = {Θ−1 ◦ G(ξ)}ξ∈B (0,3δ)
each α ∈ A, and G ∈ Σ
α
d1
+
e
and the enlarged leaf Wα,G
= {Θ−1
◦
G(ξ)}
.
For
each
α
∈
A,
G
∈Σ
α
p ξ∈Bd1 (0,6δ)
. −1
note that Wα,G ⊂ U α = Θα (Bd (0, 6δ 1 + ρ2 )) ⊆ Uα . Next we consider sets
e (our sets of “stable” leaves) such that, if Wα,G ∈ Σα and Wα,G ∩U β 6=
Σα ⊂ Σ
∅, then there exists G0 ∈ Σβ such that Wα,G ∩ U β ⊂ Wβ,G0 .21 In the following
we will be interested in the set of manifolds Σ = ∪α∈A ∪G∈Σα {Wα,G }.
s
b`,s
Also, for each G ∈ Σα we denote by Γ
0 (α, G) the C sections of the fiber
+
bundle on Wα,G , with fibers Λ` (T ∗ M ), which vanish in a neighborhood of
∂Wα,G . We define the norm
.
kgkΓb`,s (α,G) = sup kgα,ī ◦ Θ−1
α ◦ GkC s (Bd1 (0,2δ)) .
0
(3.8)
ī
Consistent with this choice, we equip Ω`r (M ) with the norms, for s ≤ r,
khkΩ`s =
sup khα,ī ◦ Θ−1
α kC s (Rd )
(3.9)
α∈A,ī∈I`
18
We will define the manifolds on which we will integrate by representing them as
graphs of functions, here we define the class of functions we are going to use.
19
We will often ignore the arguments ρ, L when it does not create confusion.
20
These Wα,G are not to be confused with stable manifolds.
. e
21
bα =
Such sets exist. For example, start by setting Σ
Σ(ρ− , L0 ), for some L0 > 0. Then,
e
for each α ∈ A and Wα,Gx,F , we consider F (ξ) = χ(ξ)F (ξ), where χ ∈ C ∞ , χ(ξ) = 1 if
kξk ≤ 3δ and χ(ξ) = 0 if kξk ≥ 6δ. Note that Fe can be trivially extended to Rds . Obviously
Wα,Gx,F = Wα,Gx,Fe even though they have different enlarged versions. For each β such
.
that Wβ = Wα,Gx,F ∩ U β 6= ∅, Fe determines, provided L has been chosen large enough, an
element Fβ ∈ Fr , thanks to condition (3.3), and xβ = (ξβ , 0) such that Wβ,Gxβ ,Fβ ⊃ Wβ .
b β . Note that the added elements automatically have the
We add such elements to Σ
required property, thanks to (3.3), since they correspond to manifolds with tangent spaces
in the cones in Uα .
18
3.2 Banach Spaces
3 CONES AND BANACH SPACES
b s (α, G) be the set of C s (Uα,G ) vector fields, where Uα,G is any open set
Let V
+
such that Uα ⊃ Uα,G ⊃ Wα,G
.
`,s
s
b`,s
Next, we choose sets Γ0 (α, G) that are dense in Γ
0 (α, G) in the C norm
b s (α, G) with the property
for each s < r. Also we choose sets V s (α, G) ⊆ V
that each of them is a Lie subalgebra and it contains the push forward of any
constant vector field under the coordinate map Θ−1
α .
Remark 3.1. The norms and the Banach spaces we are going to define
should have an index specifying their dependencies on the choices of {Γ`,s
0 (α, G)}
s
and {V (α, G)}. We choose to suppress them to ease notation, since this creates no confusion.
Let ωvol be the volume form induced on Wα,G by the Riemannian structure
of M . Write Lv for the Lie derivative along a vector field v. Finally, for all
s
`
α ∈ A, G ∈ Σα , g ∈ Γ`,0
0 (α, G), v1 , . . . , vp ∈ V (α, G) and h ∈ Ωr (M ) we
define
Z
.
(3.10)
hg, Lv1 · · · Lvp hi ωvol ∈ R.
Jα,G,g,v1 ,...,vp (h) =
Wα,G
Next, for all p ∈ N, q ∈ R+ , p + q < r − 1, ` ∈ {0, . . . , d}, let
.
, vj ∈ V p+q ,
Up,q,` = Jα,G,g,v1 ,...,vp α ∈ A, G ∈ Σα , g ∈ Γ`,p+q
0
kgkΓ`,p+q (α,G) ≤ 1, kvj kC q+p (Uα,G ) ≤ 1 .
(3.11)
0
Lastly, for all p ∈ N, q ∈ R+ and h ∈ Ω`p+q (M ) we define the following norms
.
khk−
p,q,` =
sup J(h)
and
J ∈ Up,q,`
.
khkp,q,` = sup khk−
n,q,` .
(3.12)
n≤p
For all p ∈ N, q ∈ R+ , ` ∈ {0, . . . , d − 1} we define the spaces B p,q,` to be the
closures of Ω`0,r (M ) with respect to the norm k·kp,q,` .
Remark 3.2. For the norms just defined we trivially have
khkp,q,` ≤ khkp+1,q,` and
sup
khk−
p,q,` =
khkp,q+1,` ≤ khkp,q,`
kLv1 · · · Lvp hk−
0,p+q
{v1 ,...,vp ∈V p+q : kvj kC p+q ≤1}
khkp,q,` ≤ C# khkΩ`p .
19
(3.13)
3.2 Banach Spaces
3 CONES AND BANACH SPACES
Remark 3.3. The above spaces are the natural extensions of the spaces B p,q
in [19] to the case of `-forms. There the Banach spaces B p,q were defined as
the closure of C ∞ (M, R) with respect to the following norm22
Z
sup
khkp,q = sup sup
sup
Lv1 · · · Lvk (h) · ϕωvol .
0≤k≤p α∈A v1 ,...,vk ∈V(Wα,G ) ϕ∈C q (Wα,G ,R)
0
G∈Σ̃
|vi |C r ≤1
|ϕ|C k+q ≤1
Wα,G
In particular, we can construct an isomorphism between the Banach space
B p,q in [19] and the present Banach space B p,q,d−1 . In fact, let iV : Ω`+1
r (M ) →
`
Ω0,r (M ) be the interior product defined by iV (h)(v1 , . . . , v` ) = h(V, v1 , . . . , v` ).
If h ∈ Ω`0,r (M ), then (−1)` iV (h∧dV ) = h where dV is any one form such that
`+1
`
dV (V ) = 1. That is iV (Ω`+1
r (M )) = Ω0,r (M ). Next, since iV (Ω0,r (M )) = 0,
`+1
`
we can define iV,0 : Ω`+1
r (M )/Ω0,r (M ) → Ω0,r (M ) and obtain by a standard
algebraic construction that iV,0 is a natural isomorphism. Thus we can define
.
ω
e = iV ω,
(3.14)
where ω is the Riemannian volume. Now we define i : C r (M ) → Ωd−1
0,r (M )
.
d
where i(f ) = f · ω
e , which is an isomorphism since Ω0,r = {0}. Clearly i can
be extended to an isomorphism between B p,q and B p,q,d−1 .
We now prove some properties of the spaces B p,q,` . To this end we will
use some estimates on how fast an element of the space can be approximated
by smooth forms. Such estimates will be proven in Subsection 3.3.
Given a form h ∈ Ω` , we can define a functional by
.
[(h)](g) = hh, giΩ`
where
g ∈ Ω`s (M ).
(3.15)
The space of such functionals, equipped with the ∗-weak topology of Ω`s (M )0 ,23
gives rise to the space Es` of currents of regularity s.
The following extends [19, Proposition 4.1] and [19, Lemma 2.1].
Lemma 3.4. For each ` ∈ {0, . . . , d − 1}, there is a canonical injection from
`
the space B p,q,` to a subspace of Ep+q
.
22
In [19] the coordinate charts are chosen with slightly different properties. However
for our purposes they are equivalent.
23
As usual, given a Banach space B, by B 0 we mean the dual space.
20
3.2 Banach Spaces
3 CONES AND BANACH SPACES
Proof. Since we can foliate M by manifolds in Σ, given the definition (3.15)
and equation 3.4 we have
[(h)](g) ≤ C# khkp,q,` kgkΩ`p+q (M ) .
(3.16)
`
Thus  can be extended to a continuous immersion of B p,q,` in Ep+q
.
`
To show injectivity consider a sequence {hn } ⊂ Ω0,p+q that converges to
h in B p,q,` such that (h) = 0. By Lemma 3.7 and (3.21) we have that
Z
Z
hg, hn iωvol =
limhg, Mε hn iωvol = lim[(hn )](gε ),
Wα,G ε→0
Wα,G
ε→0
where
. X
gε (x) =
ωβ,i (x)JΘβ (x)
Z
ωvol (y) ψβ (y)κ(Θβ (x) − Θβ (y))hωβ,i , giy ∈ Ω`r .
Wα,G
β,ī
Moreover, by equation (3.16) and Lemma 3.7 it follows that
|[(hn )](gε ) − [(hm )](gε )| ≤ C# kgkΓ`,q (α,G) khn − hm k0,q,` .
0
Thus, we can exchange the limits with respect to n and ε to obtain
Z
Z
hg, hiωvol = lim
hg, hn iωvol
Wα,G
n→∞
Wα,G
= lim lim [(hn )](gε ) = lim[(h)](gε ) = 0.
ε→0 n→∞
ε→0
which implies khk0,q,` = 0. By similar computations, using equations (3.20)
and (3.23) to deal with the derivatives, we obtain khkp,q,` = 0. Thus  is
injective and we obtain the statement of the theorem.
We conclude this section by proving a compactness result which is essential in the implementation of the usual Lasota-Yorke strategy.
Lemma 3.5. For each q > 0, p ≥ 1, the unit ball of B p,q,` is compact in
B p−1,q+1,` with respect to the norm k · kp−1,q+1,` .
Proof. It suffices to prove sequential compactness. Let {hn } ⊂ Bp,q,` , khn kp,q,` ≤
1. By Lemma 3.7 we can construct {hε,n } such that khε,n −hn kp−1,q+1,` ≤ C# ε
−ds −q
and khε,n
. By the Ascoli-Arzelá theorem there exists a subse kC p ≤
C# ε
quence hε,nj that is convergent in C p−1 . Such a sequence is also convergent
in B p−1,q+1,` . Compactness follows then by the usual diagonalization argument.
21
3.3
3.3
Averaging operators
3 CONES AND BANACH SPACES
Averaging operators
The following operators are used extensively in this paper.
R
Definition 3.6. Let κ ∈ C ∞ (Rd , R+ ) such that κ(x)dx = 1, supp κ ⊂ {x ∈
.
Rd : kxk ≤ 1} and let κε (x) = ε−d κ(ε−1 x). For each α ∈ A, ` ∈ {0, . . . , d},
let Ψ ∈ C0∞ (R2d , GL( d` , R)) with supp Ψ ⊂ Bd (0, 2δ)2 . We define24 the
operator Mα,Ψ,ε : Ω`r → Ω`r by

if x 6∈ Uα
 0P
R
.
κ
(Θ
(x)
−
y)
Ψ(Θ
(x),
y)
Mα,Ψ,ε (h)(x) =
α
α
d
ī,j̄ ε
 ī,j̄∈I` R
×hωα,j̄ , hiΘ−1
ωα,ī
if x ∈ Uα .
dy
α (y)
.
Given the particular choice Ψ(x, y)ī,j̄ = ψα (Θ−1
α (x))δī,j̄ we define
. X
. X
Mα,Ψ,ε .
Mα,ε =
Mε =
α
α
Lastly, given the duality induced by (3.4), we define the operators M0α,ε , M0ε ,
as those such that for all h, g ∈ Ω`r we have25
hM0α,ε h, giΩ`r = hh, Mα,ε giΩ`r and
hM0ε h, giΩ`r = hh, Mε giΩ`r .
Lemma 3.7. There exists ε0 > 0 such that, for each ε ∈ (0, ε0 ), ` ∈
{1, . . . , d}, q > 0, p ∈ N∗ and h ∈ Ω`p+q , we can bound
kMε hkp,q,` + kM0ε hkp,q,` ≤ Cp,q,κ khkp,q,` ,
kMε hkC p (Ω,M ) + kM0ε hkC p (Ω,M ) ≤ Cp,q,κ ε−ds −q khkp,q,` ,
kh − Mε hkp−1,q+1,` + kh − M0ε hkp−1,q+1,` ≤ Cp,q,κ εkhkp,q,` .
Proof. We will give a proof only for the operator Mε , the proof for M0ε being
similar. To prove the first inequality we begin by estimating the integral in
(3.10) for the case p = 0. First of all note that, for each α, β ∈ A, Wβ,G ∈ Σ
and Ψ, if
Z
hg, Mα,Ψ,ε hiωvol 6= 0,
(3.17)
Wβ,G
24
Obviously our definition depends on the choice of the function κ, and in the future
some special choice of κ will be made. However, we choose not to explicitly show the
dependence as a subscript to simplify the notation.
25
By the duality relation M0 is then also defined on currents. In addition, note that
Mα,ε Ω`0,r ∪ M0α,ε Ω`0,r ⊂ Ω`0,r .
22
3.3
Averaging operators
3 CONES AND BANACH SPACES
then Wβ,G ∩ U α 6= ∅. By the definition of Σ, it follows that there exists
Wα,G0 ∈ Σ such that Wβ,G ∩ U α ⊂ Wα,G0 . On the other hand, by Definition
3.6, the integrand is supported in U α , thus
Z
Z
hg, Mα,Ψ,ε hiωvol =
hg, Mα,Ψ,ε hiωvol .
Wβ,G
Wα,G0
Thus is suffices to compute
Z
X Z
hg, Mα,Ψ,ε hi ωvol =
Wα,Gx,F
Z
dξ
Bds (0,2δ)
ī,j̄∈I`
dy Ψ(Gx,F (ξ), y)ī,j̄
Rd
−1
× gα,ī ◦ Θ−1
α ◦ Gx,F (ξ)hα,j̄ ◦ Θα (y)κε (Gx,F (ξ) − y),
where hα,ī (z) = hh, ωα,ī iz and gα,ī (z) = hg, ωα,ī iz for each z ∈ Uα .
Next, we keep ξ ∈ Bds (0, 2δ) fixed and we consider the change of variables
ζ = Gx,F (ξ) − y. By Fubini-Tonelli theorem and the definition of Gx,F we
can exchange the integrals and obtain
Z
Z
X Z
dξ Ψ(Gx,F (ξ), Gx−ζ,F (ξ))ī,j̄
hg, Mα,Ψ,ε hiωvol =
dζ κε (ζ)
Wα,Gx,F
ī,j̄∈I`
Rd
Bds (0,2δ)
−1
× gα,ī ◦ Θ−1
α ◦ Gx,F (ξ) hα,j̄ ◦ Θα ◦ Gx−ζ,F (ξ).
(3.18)
Now given our admissible test function g and a leaf Wα,Gx,F , we can define
.
admissible elements gζ,Ψ and Wα,Gxζ ,F , as follows. Let xζ = x − ζ, gζ,Ψ,j̄ ◦
P
P
.
−1
Θ−1
α (z) =
ī gζ,Ψ,ī ωα,ī , so that
ī Ψ(z + ζ, z)ī,j̄ gα,ī ◦ Θα (z + ζ) and gζ,Ψ =
X
−1
Ψ(Gx,F (ξ), Gx−ζ,F (ξ))ī,j̄ gα,ī ◦ Θ−1
α ◦ Gx,F (ξ) = gζ,Ψ,j̄ ◦ Θα ◦ Gxζ ,F (ξ).
ī
Then
Z
Z
Z
hg, Mα,Ψ,ε hiωvol =
hgζ,Ψ , hiωvol .
dζκε (ζ)
Rd
Wα,Gx,F
Wα,Gx
(3.19)
ζ ,F
Note that, since the cones are constant in the charts, Gxζ ,F ∈ Σ for each ζ
small enough. Since, kgζ,Ψ kΓ`,q (α,Gx ,F ) ≤ Cq kΨkC q kgkΓ`,p+q (α,Gx,F ) we have
0
0
ζ
kMα,Ψ,ε hk0,q ≤ Cq kΨkC q khk0,q .
23
3.3
Averaging operators
3 CONES AND BANACH SPACES
The first inequality of the Lemma, for p = 0, follows.
Next we treat the case p > 0. We need to compute
P Lv Mα,Ψ,ε . To do so,
by using the product rule, recall that Lv (ωα,ī ) = j̄ ρī,j̄ (v)ωα,j̄ . Moreover
.
by equation (A.6) there exists Ψ0v such that we have Lv (Ψ(Θα (x), y))ī,j̄ =
. Lastly, we deal
Ψ0v (Θα (x), y))ī,j̄ . Next, we apply (A.5) to Lv hωα,j̄ , hiΘ−1
α (y)
with Lv (κε (Θα (x) − y)) using integration by parts with respect to y.
By using all the remarks above, we obtain that there existsPvectors
w1 (v), . . . , wl (v), and functions Ψv , Ψv,1 , . . . , Ψv,l such that kΨv kC r + l kΨv,l kC r ≤
C# kψkC r and for which we have
X
Lv Mα,Ψ,ε h = Mα,Ψv ,ε h +
Mα,Ψv,l ,ε (Lwl h).
(3.20)
l
By iterating the above equality, using the previous bound for p = 0 and
equation (3.13), we obtain the first inequality of the Lemma.
To prove the second inequality, given definition (3.9), we must evaluate
Z
−1
hωα,i , Mα,ε hiΘ−1
κε (z − y)hωα,ī , hiΘ−1
= ψα ◦ Θα (z)
dy
α (z)
α (y)
Rd
Z
Z
(3.21)
dyds +1 . . . dyd
hgε,z,y , hiωvol .
=
Rdu +1
Wα,G(0,y
ds +1 ,...,yd ),0
= J[Θα ◦ G(0,yds +1 ,...,yd ),0 ](ξ)−1 κε (z − (ξ, yds +1 , . . . , yd ))
where
and gε,z,y = ĝε,z,y,ī ωα,ī . Then, differentiating with respect to z, yields the
desired result.
Finally note that by construction
. b
lim[Mα,Ψ,ε h](x) = Ψ(Θα (x), Θα (x))h(x) = Ψ
α (x)h(x).
26
ĝε,z,y,ī ◦ Θ−1
α (ξ)
ε→0
Thus, by (3.20), we have
b α h].
lim Lv Mα,Ψ,ε h = Lv [Ψ
ε→0
(3.22)
Hence, by (3.19) and the explicit representation of (3.18), we can write
Z
Z 1
Z
h
iZ
d
b
ωvol hg, Ψα hi − hg, Mα,Ψ,ε hi = dζκε (ζ) dt
ωvol hgtζ,Ψ , hi
Wα,Gx,F
0 dt Wα,Gx ,F
tζ
Z
Z 1 Z
X
= dζκε (ζ)
dt
ωvol hgtζ,Ψ , Lvζ hi −
hgtζ,Ψ , ĥα,ī Lvζ ωα,ī i
0
Wα,Gx
ī
tζ ,F
(3.23)
26
As in (A.4), J[f ] is the Jacobian of f .
24
4 TRANSFER OPERATORS
where vζ = −(Θ−1
α )∗ ζ. The last inequality, for p = 0, follows since kvζ kC p+q ≤
C# ε. To obtain the result for p > 0, on can use (3.20) and (3.22) repeatedly.
The above Lemma has a useful corollary.
Corollary 3.8. Both Mε and M0ε extend to bounded operators on B p,q,` and
0
Bp,q,`
, respectively. In addition, if h ∈ B p,q,` , then
lim kh − Mε hkp,q,` + kh − M0ε hkp,q,` = 0.
ε→0
(3.24)
Proof. By definition, there exists a sequence {hn } ⊂ Ω`s that converges to h
in B p,q,` . Hence
kh − Mε hkp,q,` ≤ kh − hn kp,q,` + khn − Mε hn kp,q,`
≤ kh − hn kp,q,` + C# εkhn kp+1,q−1,`
implies the result.
4
Transfer operators and Resolvents
Let φt : M → M be a C r Anosov flow on a smooth Riemannian d-dimensional
compact manifold with r ≥ 2. The flow induces canonically an action φ∗t on
`-forms. Such an action has nice spectral properties only with respect to
carefully constructed norms of the type described in section 3. Accordingly,
we must specify all the choices involved in the definition of the norms k·kp,q,` .
First of all, the “special” direction is obviously given by the vector field
V generating the flow. In addition, we note the following.
Remark 4.1. Without loss of generality, we can assume

du
u
−1
 D0 Θ−1
α {(0, u, 0) : u ∈ R } = E (Θα (0))
D0 Θ−1 {(s, 0, 0) : s ∈ Rds } = E s (Θ−1
α (0))
 −1 α
−1
Θα ((s, u, t)) = φt Θα ((s, u, 0)).
(4.1)
Let d1 = ds and d2 = du + 1. Given the continuity of the stable and
unstable distribution we can choose δ so that equations (3.2) and (3.3) are
25
4.1 Properties of the transfer operator
4 TRANSFER OPERATORS
satisfied with ρ− = 1, ρ = 2. Moreover, for all α ∈ A and x ∈ Uα we have
(Θα )∗ E s (x) ⊂ C 1 . Note that there exists t0 > 0 such that, for all t ≥ t0 ,27
2
(Θβ ◦ φ−t ◦ Θ−1
α )∗ (C2 ) ⊂ C1
(4.2)
λt
and for each v ∈ (Θ−1
α )∗ C2 , t ∈ R+ it holds k(φ−t )∗ (v)k > C# e kvk. We
b`q,0 .
b s and Γ`q,0 = Γ
define Σ as in footnote 21 for L large enough, V s = V
This concludes the definition of the Banach spaces and next we must
define the action of the flow.
4.1
Properties of the transfer operator
Let h ∈ Ω`0,s (M ), then ∀t ∈ R
φ∗t h(V, v2 , . . . , v` ) = h(V, (φt )∗ v2 , . . . , (φt )∗ v` ) = 0.
(4.3)
Thus φ∗t (Ω`0,s ) = Ω`0,s for all t ∈ R. We recall (2.7), where we defined the
(`)
operators Lt : Ω`0,s (M ) → Ω`0,s (M ), t ∈ R+ ,
.
(`)
Lt h = φ∗−t h.
Following the remark (3.3), in the special case of a d − 1 form, for h ∈
we have h = h̄e
ω where h̄ ∈ C s (M, R) and ω
e is defined in (3.14).
Then
.
(d−1)
Lt
h = h̄ ◦ φ−t det(Dφ−t )e
ω = (Lt h̄)e
ω.
(4.4)
Ωd−1
0,s (Uα )
28
Thus we recover the transfer operator acting on densities studied in [10].
(`)
We begin with a Lasota-Yorke type inequality for Lt .
Definition 4.2. Let p ∈ N, q ∈ R+ , 0 ≤ ` ≤ d − 1 and 0 < λ < λ where λ
is as in the Anosov splitting. Let us define
.
σ` = htop (φ1 ) − λ|ds − `|
.
σp,q = min{p, q}λ .
27
Indeed, if x ∈ Uα , v ∈ Rd and v = v u + v s , v u ∈ (Θα )∗ (E u (x) × E c (x)), v s ∈
(Θα )∗ (E s (x)), then k((φ−t ◦ Θ−1 )∗ (v u )k ≤ C0 while k((φ−t ◦ Θ−1 )∗ (v s )k ≥ C0 eλt and the
result follows by the fourth part of (3.1).
28
∗
Note that for ω ∈ Ωd−1
0,s (M ) we have det(Dφt )ω = φt ω. Hence by (4.3) we have
∗
∗
φt ω
e (v1 , . . . , vd−1 ) = det(Dφt )e
ω (v1 , . . . , vd−1 ). That is φt ω
e = det(Dφt )e
ω.
26
4.1 Properties of the transfer operator
4 TRANSFER OPERATORS
Lemma 4.3. There exists t0 > 0 such that, for each p, q ≥ 0, p + q < r − 1,
(`)
` ∈ {0, . . . , d−1} and t > t0 the linear operators Lt are bounded in the k·kp,q,`
(`)
norm. Accordingly, each such Lt can be uniquely extended to a bounded
(`)
operator29 Lt : B p,q,` → B p,q,` . Moreover,
(`) ≤ Cq eσ` t khk0,q,` .
Lt h
0,q,`
In addition, if p, q > 0,
(`) ≤Cq,λ e(σ` −λq)t khk0,q,` + Cq,λ eσ` t khk0,q+1,`
Lt h
0,q,`
(`) ≤Cp,q,λ e(σ` −σp,q )t khkp,q,` + Cp,q,λ eσ` t khkp−1,q+1,`
Lt h
p,q,`
+ Cp,q,λ eσ` t X (`) hp−1,q+1,` .
The proof of this lemma is the content of subsection 4.3. In particular,
note that from the first and last equations of the above lemma and equation
(3.13) we have for all p, q ≥ 0,
(`) ≤ Cp,q eσ` t khkp,q,` .
Lt h
p,q,`
Remark 4.4. Up to now we have extended, yet followed closely, the arguments in [10]. Unfortunately, in Section 7 of [10], the authors did not take
into consideration that for t ≤ t0 , φt (Wα,G ) is not necessarily controlled by
the cones defined by (3.2), and thus could be an inadmissible manifold. Such
an issue can be easily fixed by introducing a dynamical norm (similarly to
[5]) as done below.
To take care of the t ≤ t0 , we introduce the dynamical norm |||·|||p,q,` . For
each h ∈ Ω`r (M ), we set
.
|||h|||p,q,` = sup L(`)
s h p,q,` .
s≤t0
|||·|||p,q,`
.
Thus we can define a space Bep,q,` = Ω`0,r
⊂ B p,q,` .
29
(`)
Which, by a harmless abuse of notation, we still designate by Lt .
27
4.2 Properties of the Resolvent
4 TRANSFER OPERATORS
Lemma 4.5. For each p ∈ N, q ∈ R, p, q > 0, p+q < r−1, ` ∈ {0, . . . , d−1},
(`)
λ < λ, for each t ∈ R+ , we have that Lt ∈ L(Bep,q,` , Bep,q,` ). Moreover
(`) ≤Cq eσ` t |||h|||0,q,`
Lt h
0,q,`
(`) ≤Cp,q,λ e(σ` −σp,q )t |||h|||p,q,` + Cp,q,λ eσ` t |||h|||p−1,q+1,`
Lt h
p,q,`
+ Cp,q eσ` t X (`) hp−1,q+1,` .
Moreover, {Lt }t∈R+ forms a strongly continuous semigroup.
Proof. For h ∈ Ω`0,r Lemma 4.3 implies, for t < t0 ,
n
o
(`) ≤ max |||h|||0,q,` , Cq e|σ` |t0 |||h|||0,q,` ≤ Cq |||h|||0,q,` ,
Lt h
0,q,`
and, for t ≥ t0 , the inequality holds trivially. A similar argument implies
the second inequality. The estimate for p > 0 is obtained by induction. The
(`)
boundedness of Lt follows.
(`)
To conclude, note that Lt is strongly continuous on Ω`r (M ) in the C r
topology by equation (3.13). Let {hn } ⊂ Ω`0,p such that limn→∞ |||hn − h|||p,q,` =
(`)
0, then, by the boundedness of Lt ,
(`)
(`)
L
h
−
h
≤ C# |||h − hn |||p,q,` + kLt hn − hn kΩ`p
t
p,q,`
which can be made arbitrarily small by choosing first n large and then t
small.
4.2
Properties of the Resolvent
(`)
By standard results, see for example [12], the semigroups Lt have generators
(`)
(`)
X (`) which are closed operators on Bep,q,` such that X (`) Lt = dtd Lt . If we
compute X (`) R(`) (z)n and R(`) (z)n X (`) we obtain the following identity
Z ∞
1
(`)
(`)
n
tn−1 e−zt Lt dt.
(4.5)
R (z) =
(n − 1)! 0
The next Lemma, which proof can be found at the end of section 4.3, gives
an effective Lasota-Yorke inequality for the resolvent.
28
4.2 Properties of the Resolvent
4 TRANSFER OPERATORS
Lemma 4.6. Let p ∈ N, q ∈ R, p + q < r − 1, z ∈ C such that a = <(z) > σ`
we have
(`) n R (z) ≤ Cp,q,λ (a − σ` )−n
p,q,`
(`) n (|z| + 1)
−n
R (z) h
≤ Cp,q,a,λ (a − σ` + σp,q ) |||h|||p,q,` +
|||h|||p−1,q+1,` .
p,q,`
(a − σ` )n
Moreover the operator R(`) (z) has essential spectral radius bounded by (a −
σ` + σp,q )−1 .
We are now in the position to obtain the wanted spectral properties.
Lemma 4.7. For p + q < r − 1 the spectrum of the generator X (`) of the
(`)
semigroup Lt acting on Bep,q,` lies on the left of the line {σ` + ib}b∈R and in
the strip σ` ≥ <(z) > σ` − σp,q consists of isolated eigenvalues of finite multiplicity. Moreover, for ` = ds , htop (φ1 ) is an eigenvalue of X. If the flow is
topologically transitive htop (φ1 ) is a simple eigenvalue. If the flow is topologically mixing, then htop (φ1 ) is the only eigenvalue on the line {htop (φ1 )+ib}b∈R .
Proof. The first part of the Lemma is proven exactly as in [30, Proposition
2.10, Corollary 2.11]. Next, let us analyze the case ` = ds . In each E s (x)
let ω̃s be the volume form on E s normalized so that kω̃s k = 1. Also let
πs (x) : Tx M → E s (x) be the projections on s (x) along E u (x) ⊕ E c (x). Re.
member that πs is $-Hölder (see Appendix D). Next, define ωs (v1 , . . . , vds ) =
s
. Note that φ∗−t ωs = Js φ−t ωs ,
ω̃s (πs v1 , . . . , πs vds ), by construction ωs ∈ Ωd0,$
where Js φ−t is the Jacobian restricted to the stable manifold. Finally, consider
Z
Z
Z
hωs , Lt ωs i =
Js φ−t ≥ C#
JW φ−t ≥ C# vol(φ−t Wα,G ). (4.6)
Wα,G
Wα,G
Wα,G
Taking the sup on the manifolds and integrating in time, Appendix C (see in
particular Remark C.4) implies that the spectral radius of Rds (a) on Be0,$,ds
is exactly (a − σds )−1 . Next, the third equation in the statement of Lemma
4.3 implies that the spectral radius of Rds (a) is (a − σds )−1 when acting on
each Be0,q,ds , q ≥ 0. In addition, since khk0,q,ds ≤ khkC 0 , it follows that
kMε h − hk0,q,ds ≤ C# ε$ . Thus
d
R s (a)n Mε h
p,q,ds
C0,q,ds ,λ [1 − C0,q,ds ,λ ε$ ]
. (4.7)
≥ Rds (a)n Mε h0,q,ds ≥
(a − σds )n
29
4.3 Proof of the Lasota-Yorke
4 TRANSFER OPERATORS
Accordingly, the spectral radius of Rds (a) is (a − σds )−1 when acting on each
Bep,q,ds . Lemma 4.6 implies then that the operator is quasicompact and that
its peripheral spectrum does not contain Jordan blocks.30 In turns, this
implies that
(
n−1
Π if (a − σds )−1 ∈ σBep,q,ds (Rds (a))
1X
(a − σds )k Rds (a)k =
lim
n→∞ n
0
otherwise,
k=0
where Π is the eigenprojector on the associated eigenspace and the convergence takes place in the strong
of L(Bep,q,ds , Bep,q,ds ). ComR operator topology
puting, for ε small enough, Wα,G hMε ωs , Rds (a)k Mε ωs i as in (4.6), (4.7) it follows that 0 < kΠMε ωs k0,q,ds ≤ kΠMε ωs kp,q,ds . Hence Π 6= 0 and (a − σds )−1
belong to the spectrum. The other claimed properties follow by arguing as
in [20, Section 6.2].
4.3
Proof of the Lasota-Yorke inequality
Proof of Lemma 4.3. For each a ∈ (0, 1), we introduce the norms
p
khkp,q,`,a
. X ℘
=
a khk−
℘,q,` .
℘=0
As the above norms are equivalent to k · kp,q,` , it suffices to prove the required
inequality for some fixed a (to be chosen later).
e vi ∈ V ℘+q (W + ), g ∈ Γ0`,℘+q (α, G) such
For any ℘, q ∈ N, α ∈ A, G ∈ Σ,
α,G
that kgkΓ`,℘+q (α,G) ≤ 1, we must estimate
0
Z
(`)
hg, Lv1 · · · Lv℘ Lt hi ωvol .
(4.8)
Wα,G
First of all, we consider the atlas introduced at the beginning of section 3.
For β ∈ A and for each t ∈ R+ , let {Vk,β }k∈Kβ , Kβ ⊂ N, be the collection
.
eβ =
of connected components of Θβ (φ−t (Wα,G ) ∩ Uβ ). Let K
{k ∈ Kβ :
e
Vk,β ∩ B(0, 2δ) 6= ∅}. For all Vk,β , let Vk,β be the connected component of
+
e β choose
Θβ (φ−t (Wα,G
) ∩ Uβ ) which contains Vk,β . Next, for each k ∈ K
A Jordan block would imply that Rds (a)n p,q,ds grows at least as C# n(a − σds )−n
contrary to the first inequality in Lemma 4.6.
30
30
4.3 Proof of the Lasota-Yorke
4 TRANSFER OPERATORS
xk,β ∈ (Vk,β ∩ B(0, 2δ)). By standard uniform hyperbolicity estimates it
follows that, provided t > t0 , there exists Fk ∈ Fr (2, L) (see definition (3.7))
such that Gxk,β ,Fk,β (ξ) = xk,β + (ξ, Fk,β (ξ)), ξ ∈ B(0, 6δ), and the graph of
Gxk,β ,Fk,β is contained in Vek,β . Next we consider the manifolds
{Wβ,Gxk,β ,Fk,β }k∈K̃β ⊂ Uβ .
(4.9)
From now on let us write Wβ,Gk for Wβ,Gxk,β ,Fk,β . Note that the above construction provides an explicit analogue of Lemma 7.2 in [10].
In the following it will be convenient to use the Hodge operator “∗”,
see Appendix A for details. Let us start with the case ℘ =
0. By using
31
formula (A.4), changing variables and setting JW φt = det Dφt |T Wβ,G ,
k
we obtain32
Z
XZ
(`)
hg, Lt hiωvol =
Wα,G
β∈A
eβ
k∈K
(−1)(d−`)` ψβ
Wβ,Gk
JW φt
h∗φ∗t (∗g), hiωvol .
Jφt
(4.10)
Note that
∗φ∗t (∗g) =
X
hωβ,ī , ∗φ∗t ∗ωα,j̄ igj̄ ◦ φt · ωβ,ī
ī,j̄
= (−1)(d−`)` Jφt
X
hφ∗−t ωβ,ī , ωα,j̄ i ◦ φt · gj̄ ◦ φt · ωβ,ī ,
ī,j̄
where we have used (A.4) again. Accordingly, we have
−1
−1
∗
k[Jφ−1
t JW φt · ψβ · (∗φt (∗g))] ◦ Θβ ◦ Gk kC q ≤ C# sup kgj̄ ◦ φt ◦ Θβ ◦ Gk kC q
j̄
× k[JW φt ·
hφ∗−t ωβ,ī , ωα,j̄ i
◦ φt ] ◦
Θ−1
β
◦ Gk kC q .
(4.11)
Next, we estimate the factors of the above product. Note that there exist
maps Ξt ∈ C r (Rd , Rd ) such that
Θα ◦ φt ◦ Θ−1
β ◦ Gk = G ◦ Ξt .
31
(4.12)
In the following will use interchangeably the notations det(Dφt ) and Jφt , since, for
all t ∈ R, det(Dφt ) > 0.
32
Since hg, hi can be seen as a function on M , we will often use the notation F ∗ hg, hi
for hg, hi ◦ F .
31
4.3 Proof of the Lasota-Yorke
4 TRANSFER OPERATORS
Moreover, by the contraction properties of the Anosov flow, kDΞt kC r−1 ≤
C# e−λt .33 Thus, by the properties of r-norms (see footnote 17),
−1
kgj̄ ◦ φt ◦ Θ−1
β ◦ Gk kC q ≤ C# kg ◦ Θα ◦ GkC q .
(4.13)
Next, note that |hφ∗−t ωβ,ī , ωα,j̄ i| is bounded by the growth of `-volumes while
JW φt gives the contraction of ds volumes in the stable direction. Clearly, the
latter is simply the inverse of the maximal growth of |hφ∗−t ωβ,ī , ωα,j̄ i| which
takes place for ds forms. By the Anosov property it follows that
−1
∗
−|ds −`|λt
kJW φt ◦ Θ−1
. (4.14)
β ◦ Gk · hφ−t ωβ,ī , ωα,j̄ i ◦ φt ◦ Θβ ◦ Gk kC 0 ≤ C# e
To compute the C q norm, for q ≥ 1, we begin by computing the Lie derivative
.
with respect to the vector fields Zi = ∂ξi , i ∈ {1, . . . , ds }. Let us set Υt =
φt ◦ Θ−1
β ◦ Gk . By (A.5) and (A.6) we have
LZi [hφ∗−t v, wi ◦ Υt ] = Υ∗t LΥt∗ Zi hφ∗−t v, wi
= −Υ∗t [div(Υt∗ Zi )h(φ∗−t v, wi] + (−1)`(d−`) Υ∗t h∗LΥt∗ Zi ∗w, φ∗−t vi (4.15)
+ Υ∗t hw, L(Θ−1 ◦Gk )∗ Zi vi.
β
Hence, recalling again (3.6) and its properties, for q ∈ N,
khφ∗−t v, wi ◦ Υt kC q ≤ C# sup kLZi [hφ∗−t v, wi ◦ Υt ]kC q−1
i
+ 2bqc+1 khφ∗−t v, wi ◦ Υt kC 0
≤ Cq k div(Υt∗ Zi )hφ∗−t v, wikC q−1 + khL(Θ−1 ◦Gk )∗ Zi v, wikC q−1
β
∗
∗
+ khφ−t v, ∗LΥt∗ Zi ∗ wikC q−1 + khφ−t v, wikC 0 ,
which can be used inductively to show that the C q norm is bounded by the
C 0 norm. The computation for the case q ∈ R being similar. Finally, by
(4.14), (4.13), and (4.11) we have that
−λ|ds −`|t
.
kψβ Jφ−1
t JW φ−t ∗ φt ∗ gkΓ`,q ≤ C# e
(4.16)
0
This provides an estimate for each term in (4.10). We are left with the task
of estimating the number of manifolds. Note that such number at time t is
bounded by a constant times the size of the manifold φ−t (Wα,G ). Moreover,
33
See the Appendix in [10] for details.
32
4.3 Proof of the Lasota-Yorke
4 TRANSFER OPERATORS
it turns out that such volumes grow proportionally to ehtop (φ1 )t (see Appendix
C). Thus
(`) −
≤ Cq e−λ|ds −`|t ehtop (φ1 )t khk−
(4.17)
Lt h
0,q,` .
0,q,`
To establish the second inequality note that, by analogy with [19, Lemma
0
6.6], for each ε > 0 there exists gε ∈ Γ`,r
0 (Wα,G ) such that, setting q =
max{0, q − 1},
kgε − gk
0
`,q 0
Γ0
≤ Cq εq−q ; kgε kΓ`,q ≤ Cq ; kgε kΓ`,q+1 ≤ Cq ε−1 .
0
0
Thus, writing g = (g − gε ) + gε we can apply all the above estimates to the
two pieces, where for gε we use q + 1 instead of q. To do so we use the
estimate (4.13) for the first piece, which after setting ψt,k = φt ◦ Θ−1
β ◦ Gk ,
34
for q 6∈ N, now reads
0
k(g − gε )j̄ ◦ ψt,k kC q ≤ Cq εq−q + sup H q−bqc LZi1 . . . LZibqc [(g − gε )j̄ ◦ ψt,k ]
i1 ,...,ibqc
≤ Cq (εmin{1,q} + e−λqt ),
the case q ∈ N being easier. Doing so yields the inequality
(`) −
−
≤ Cq e(htop (φ1 )−λ|ds −`|−λq)t khk−
Lt h
0,q,` + Cq,t khk0,q+1,` .
(4.18)
0,q,`
We are left with the case ℘ > 0. We can apply Lemma 7.4 in [10] to
decompose each v ∈ V ℘+q (φt (Wβ,Gxk ,Fk )) as v = v s + v u + v V where v s is
tangent to φt (Wβ,Gxk ,Fk ), v u is “close” to the unstable direction and v V is the
component in the flow direction. Let σ ∈ {s, u, V }℘ , then (4.8) reads
X Z
(`)
hg, Lv1σ1 · · · Lv℘σ℘ Lt hiωvol .
(4.19)
σ∈{s,u,V }℘
Wα,G
Since Lv Lw h = Lw Lv h + L[v,w] h, we can reorder the derivatives as to have
first those with respect to the stable components, then those with respect
to the unstable, and then finally those in the flow direction. By permuting
the Lie derivative in such a way we introduce extra terms with less than
℘ derivatives, and these extra terms contribute only to weaker norms. For
34
By H η (f ) we mean the Hölder constant of f , i.e. supkx−yk≤δ
33
kf (x)−f (y)k
.
kx−ykη
4.3 Proof of the Lasota-Yorke
4 TRANSFER OPERATORS
each σ ∈ {s, u, V }℘ let ps (σ) = #{i | σi = s}, pu (σ) = #{i | σi = u} and
pV (σ) = #{i | σi = V }. For each ps , pu , pV ∈ N, ps + pu + pV = ℘, let
Σp̄ = {σ : ps = ps (σ), pu = pu (σ), pV = pV (σ)}. Next, for each σ ∈ Σp̄ we
define the permutation πσ by πσ (i) < πσ (i + 1) for i 6∈ {ps , ps + pu }; σπ(i) = s
for i ∈ {1, . . . , ps }; σπ(i) = u for i ∈ {ps + 1, . . . , ps + pu } and σπ(i) = V for
i ∈ {℘ − pV + 1, . . . , ℘}. We can then bound the absolute value of (4.19) by
X
X Z
hg, Lvπs σ (1) · · · Lvπs σ (ps ) Lvπuσ (ps +1) · · · Lvπuσ (ps +pu )
ps +pu +pV =℘ σ∈Σp̄
× LvπV
σ (ps +pu +1)
Wα,G
· · · LvπV
σ
(`)
Lt hiωvol (℘)
(4.20)
−℘+1
+a
(`) C℘,q Lt h
.
℘−1,q,`,a
To simplify the notation we define35
 s
 vπσ (i) for i ≤ ps
vπuσ (i) for ps + 1 ≤ i ≤ ps + pu
ṽi =
 V
vπσ (i) for i ≥ ps + pu + 1.
We will analyze the terms of (4.20) one by one. First of all, suppose pV 6= 0,
then vi = γi V for i > ps + pu . Using equation (A.7),
Z
(`)
hg, Lṽ1 · · · Lṽ℘ Lt hiωvol Wα,G
Z
(`) (`)
(4.21)
=
hg, Lṽ1 · · · Lṽ℘−1 (γ℘ Lt X h)iωvol Wα,G
−
(`)
(`) ≤ C℘,q Lt X (`) h
+ a−℘+1 C℘,q Lt h
.
℘−1,q+1,`
℘−1,q,`,a
Next, suppose pV = 0 but ps 6= 0. By (A.5),
Z
(`)
hg, Lṽ1 · · · Lṽ℘ Lt hiωvol Wα,G
Z
(`) −
(`)
≤ Lṽ1 hg, Lṽ2 · · · Lṽ℘ Lt hiωvol + C℘,q Lt h
.
Wα,G
℘−1,q,`
35
In the following we should write ṽσ,i . However, since the following arguments are
done for each fixed σ, we drop the subscripts for ease the notation.
34
4.3 Proof of the Lasota-Yorke
4 TRANSFER OPERATORS
Then36
d(f iv ωvol ) = Lv (f ωvol ) = (−1)ds +1 (Lv f )ωvol + f d(iv ωvol ).
(`) −
and
Thus, by Stokes theorem, the integral is bounded by C℘,q Lt h
℘−1,q+1,`
Z
(`) −
(`)
.
(4.22)
hg, Lṽ1 · · · Lṽ℘ Lt hiωvol ≤C℘,q Lt h
Wα,G
℘−1,q,`
Finally, suppose pu = ℘. Recalling (A.6) yields
Z
Z
(`)
(`)
hg, Lṽ1 · · · Lṽ℘ Lt hiωvol = hg, Lt Lφ−t∗ ṽ1 · · · Lφ−t∗ ṽ℘ hiωvol .
Wα,G
Wα,G
By construction we have kφ−t∗ ṽi kC p+q ≤ C# e−λt . Hence, using (4.18) and
(3.13) yields for ℘ < p
Z
(`)
hg, Lṽ1 · · · Lṽ℘ Lt hiωvol ≤ Cq,t kLφ−t∗ ṽ1 · · · Lφ−t∗ ṽ℘ hk−
0,q+℘+1,`
Wα,G
(4.23)
+ Cq e(htop (φ1 )−λ|ds −`|−qλ)t kLφ−t∗ ṽ1 · · · Lφ−t∗ ṽ℘ hk−
0,q+℘,`
−
≤ Cq e(htop (φ1 )−λ|ds −`|−(q+℘)λ)t khk−
℘,q,` + C℘,q,t khk℘,q+1,` .
Note that the last term can be estimated by the k · kp−1,q+1,`,a norm, except
in the case ℘ = p. In such a case we use instead (4.17) which yields, for each
℘ ≤ p,
Z
(`)
(4.24)
hg, Lṽ1 · · · Lṽ℘ Lt hiωvol ≤ Cq e(htop (φ1 )−λ|ds −`|−℘λ)t khk−
℘,q,` .
Wα,G
Collecting together (4.18), (4.21), (4.22), (4.23) and (4.24) we have, for t > t0 ,
(`)
−
kLt hkp,q,`,a ≤ Cq e(htop (φ1 )−λ|ds −`|−qλ)t khk−
0,q,` + Cq,t khk0,q+1,`
p
X
(`) (`) (`) −
℘
+ Cp,q a Lt h
+
a C℘,q Lt X h
p−1,q,`,a
+
p−1
X
℘−1,q+1,`
℘=1
(`)
a℘ C℘,q e(htop (φ1 )−λ|ds −`|−λ(q+℘))t khk−
℘,q,` + Cp,q,a,t kLt hkp−1,q+1,`,a
℘=1
+ ap Cp,q e(htop (φ1 )−λ|ds −`|−λp)t khk−
p,q,` .
36
Here we are doing differential calculus intrinsically in the manifold Wα,G .
35
4.3 Proof of the Lasota-Yorke
4 TRANSFER OPERATORS
Then
(`)
kLt hkp,q,`,a ≤ Cp,q e(htop (φ1 )−λ|ds −`|−min{p,q}λ)t khkp,q,`,a
(`)
+ Cp,q,a,t khkp−1,q+1,`,a + Cp,q,a,t kLt hkp−1,q+1,`,a
(`)
(`) + Cp,q a Lt X (`) h
+ Cp,q a Lt h
.
p−1,q+1,`,a
(4.25)
p,q,`,a
While, if we do not use (4.18) and (4.23), we have, for a ∈ (0, 1),
kLt hkp,q,`,a ≤ Cp,q e(htop (φ1 )−λ|ds −`|)t khkp,q,`,a
(`) (`) (`) + Cp,q, a Lt h
+ Cp,q a Lt X h
p−1,q+1,`,a
.
p−1,q,`,a
By induction, for which (4.17) is the initial step, it follows that, for all t ∈ R+ ,
kLt hkp,q,`,a ≤ Cp,q e(htop (φ1 )−λ|ds −`|)t khkp,q,`,a .
(4.26)
Next, we choose t∗ such that for λ < λ we have that37
Cp,q e(htop (φ1 )−λ|ds −`|)t∗ ≤ e(htop (φ1 )−λ|ds −`|)t∗
Cp,q e(htop (φ1 )−λ|ds −`|−min{p,q}λ)t∗ ≤ e(htop (φ1 )−λ|ds −`|−min{p,q}λ)t∗ .
Finally, choosing a = e− min{p,q}t∗ , from (4.25) and (4.26) we have
kLt∗ hkp,q,`,a ≤ e(htop (φ1 )−λ|ds −`|−min{p,q}λ)t∗ khkp,q,`,a
(`) (`) + Cp,q,a,t∗ khkp−1,q+1,`,a + Cp,q a Lt∗ X h
.
(4.27)
p−1,q+1,`,a
Iterating (4.26) by time steps of size t∗ , and recalling and (4.27), the Lemma
follows.
Proof of Lemma
4.5, equation
4.6.
By
the second inequality of Lemma
(`) 38
(3.13), and since X h p−1,q+1,` ≤ |||h|||p,q,` , we have
(`) n R (z) h
p,q,`
Cp,q,λ
|||h|||p,q,`
≤
(n − 1)!
Z
∞
0
dt e−at+σ` t tn−1 =
Cp,q,λ
|||h|||p,q,` .
(a − σ` )n
Thus, provided <(z) > σ` , we have that R(`) (z)n ∈ L(Bep,q,` , Bep,q,` ).
37
38
In this case Cp,q refers exactly to the constants in (4.25) and (4.26) respectively.
R∞
Pn
αk
Note that for each a > 0, α ≥ 0, we have α xn e−ax dx = e−aα k=0 n!
k! an−k+1 .
36
4.3 Proof of the Lasota-Yorke
4 TRANSFER OPERATORS
Let us introduce a truncated resolvent
Z ∞
1
.
(`)
Rn,` (z) =
tn−1 e−zt Lt .
(n − 1)! t0
Then
(`) n
R (z) − Rn,` (z)
p,q,`
t0
Cp,q,λ tn−1 e−(a−σ` )t
(n − 1)!
0
n
t0
≤ Cp,q,λ
≤ Cp,q,λ (a − σ` + σp,q )−n ,
n!
Z
≤
(4.28)
provided n ≥ t0 e(a−σ` +σp,q ).39 To prove the second inequality of the Lemma
it suffices to estimate Rn,` (z). To this end we follow the same computation
used to estimate the norm k·kp,q,` in Lemma 4.3. The only difference being
in the treatment of (4.21). Indeed, since XLt = dtd Lt , integration by parts
yields
Z
Z ∞ n−1 −zt
t
e
(`)
hγ℘ g, Lṽ1 · · · Lṽ℘−1 (Lt X (`) h)iωvol Wα,G t0 (n − 1)!
≤ C℘,q |z| khk℘−1,q+1,` .
Hence
kRn (z)hkp,q,` ≤
Cp,q,a,λ khkp,q,`
Cp,q,a,λ (|z| + 1)
+
khkp−1,q+1,` .
n
(a − σ` + σp,q )
(a − σ` )n
(4.29)
The first part of the Lemma then follows as in Lemma 4.5.
To prove the bound on the essential spectral radius of R(`) (z)n we argue
by analogy with [30, Proposition 2.10, Corollary 2.11]. Nussbaum’s formula
[37] asserts that if rn is the infimum of the r such that {R(`) (z)n h}khk≤1 can
be covered by a finite number of balls of radius r, then the essential spectral
√
radius of R(`) (z)n is given by lim inf n→∞ n rn . Let
.
.
B1 = {h ∈ Bep,q,` | |||h|||p,q,` ≤ 1} ⊂ {h ∈ B p,q,` | khkp,q,` ≤ 1} = B2 .
By Lemma 3.5, B2 is relatively compact inSB p−1,q+1,` . Thus, for each > 0
there are h1 , . . . , hN ∈ B2 such that B2 ⊆ N
i=1 U (hi ), where U (hi ) = {h ∈
39
Since n! ≥ nn e−n .
37
5 FLAT TRACES
B p−1,q+1,` | kh − hi kp−1,q+1,` < }. For h ∈ B1 ∩ U (hi ), computing as in
(4.29) we have
|||Rn (z)(h − hi )|||p,q,` ≤
Cp,q,a,λ
Cp,q,a,λ eat0 |z|
kh
−
h
k
+
.
i
p,q,`
(a − σ` + σp,q )n
(a − σ` )n
Choosing appropriately and recalling (4.28) we conclude that for each n ∈ N
the set R(`) (z)n (B1 ) can be covered by a finite number of |||·|||p,q,` − balls of
radius Cp,q,a,λ (a − σ` + σp,q )−n , which implies the statement.
5
Flat Traces
In this section we define a flat trace and we prove some of its relevant properties. Formally, for A ∈ L(B p,q,` , B p,q,` ) we would like to define this by
Z X
hωα,ī , Aδx,ī ix
(5.1)
M
α,ī
where, for x ∈ Uα , δx,ī (f ) = hωα,ī , f ix ωα,ī (x), for each f ∈ Ω`r . Unfortunately,
the δx,ī do not belong to the space B p,q,` . We are thus forced to employ an
indirect strategy. For each x ∈ M , ī ∈ I` , α ∈ A and ε > 0 small enough,
let40
JΘα (x)ψα (x)κε (Θα (x) − Θα (y)) ωα,ī (y) if y ∈ Uα
.
j,α,ī,x (y) =
(5.2)
0
if y ∈
/ Uα .
At this stage we choose a particular κ (see definition (3.6)). Let κ be
κ(x1 , . . . , xd ) = κd−1 (x1 , . . . , xd−1 )κ1 (xd ). Given an operator A ∈ L(B p,q,` , B p,q,` )
we define the flat trace by
Z X
.
[
tr (A) = lim
hωα,ī , A(j,α,ī,x )ix ,
(5.3)
→0
M
α,ī
provided the limit exists. To get a feeling for what we are doing, consider
the operator A acting on h ∈ Ω` defined by
Z
.
Ah(x) =
a(x, y)[h(y)]
(5.4)
M
40
Note that the definition is, given the freedom in the choice of κ, quite arbitrary. We
are not interested in investigating the equivalence of the various possible definitions.
38
5 FLAT TRACES
where a(x, y) ∈ L(∧` Ty∗ M, ∧` Tx∗ M ) depends continuously on x, y. A direct
computation shows that
Z
[
tr(a(x, x)).
tr (A) =
M
Also, one can verify directly that for a finite rank operator A we have tr[ (A) =
tr(A), although we will not require this fact.
Lemma 5.1. There exists C2 > 0 such that, for each s ∈ R+ , <(z) > C2 and
(`)
n ∈ N, 0 ≤ ` ≤ d − 1 we have that tr[ (R(`) (z)n Ls ) is well defined. Moreover,
tr[ (R(`) (z)n L(`)
s ) =
X χ` (τ )
1
[λ(τ ) − s]n−1 λ(τ )e−z(λ(τ )−s) ,
(n − 1)!
µ(τ )
τ ∈T (s)
.
provided s 6∈ {λ(τ )}τ ∈T for T (s) = {τ ∈ T ; λ(τ ) > s}.
Proof. By definitions (4.5), (5.2) and (5.3) we have
Z
X tn−1 e−zt
[
n
ω(x)dt
tr (R(z) Ls ) = lim
hωα,ī , φ∗−t−s jε,α,ī,x ix
ε→0 M ×R
(n − 1)!
+
α,ī
Z
X
tn−1 e−zt
= lim
ω(x)dt
JΘα (x)ψα (x)
ε→0
(n − 1)!
Uα ×R+
(5.5)
α,ī
× κε (Θα (x) − Θα (φ−t−s (x)))hωα,ī , φ∗−t−s ωα,ī ix .
The integrand is different from zero only for κε (Θα (x) − Θα (φ−t−s (x))) 6= 0,
that is for d(x, φ−t−s (x)) ≤ ε.
Next we use the shadowing theorem for flows (Bowen [8]), in the formulation explicitly given by Pilyugin in [40, Theorem 1.5.1], adapted to our case.
First of all we define the (ε0 , T )-pseudo-orbits t(t) : R → M to be a map
such that for any t0 ∈ R we have d(φt (t(t0 )), t(t + t0 )) ≤ ε0 if |t| < T .41 Then
we have the following theorem.
Theorem 5.2 ([40]). Let M be a smooth manifold and φt a C 2 Anosov flow.
There exists ε0 > 0, C3 ≥ 1 such that given a (ε, 1)-pseudo orbit t(t) with
ε < ε0 there exists a orbit τ , a point p ∈ τ and a reparametrization σ(t) such
that, for all t, s ∈ R,
d(t(t), φσ(t) (p)) ≤ C3 ε where |σ(t) − σ(s) − t + s| ≤ C3 ε|t − s|.
41
Note that we do not require t(t) to be continuous.
39
5 FLAT TRACES
Note that, given x ∈ M and t ∈ R such that d(x, φ−t−s (x)) < ε, we can
construct a periodic ε-pseudoorbit. Then, by Theorem 5.2 and by the Anosov
closing Lemma [27, Theorem 18.1.7 and Corollary 18.1.8], we have that, for
ε small enough, there exists a closed orbit τ such that d(φ−s (x), τ ) ≤ C3 ε for
all s ∈ [0, t + s] and |λ(τ ) − t − s| ≤ C3 ελ(τ ). Moreover, the above closed
orbit is unique providing > 0 is sufficiently small. Given a closed orbit τ
we define
.
Ωτ = {(x, t) ∈ M × R+ | |t − λ(τ )| ≤ εC3 λ(τ ),
d(φ−s (x), τ ) ≤ C3 ε ∀s ∈ (0, (1 + C3 ε)λ(τ ))}.
Note that, provided ε < C3−1 ε0 , all the Ωτ are disjoint. In fact if (x, t) ∈
Ωτ ∩ Ωτ 0 , then we can associate to it a (C3 ε, 1)-pseudo orbit. Then, by
Theorem 5.2 there is only one periodic orbit with period close to λ(τ ) in a
C32 ε neighborhood of x, which implies τ = τ 0 . For further use we set
tn−1 −zt
e ψα (x)hωα,ī , φ∗−t−s ωα,ī ix ,
(n − 1)!
.
Tε (s) = T ((1 − C3 ε)s).
.
Fα,ī,z (x, t) =
We can rewrite the integral in (5.5) as
XZ
κε (Θα (x) − Θα (φ−t (x))JΘα (x)Fα,ī,z (x, t − s)
α,ī
Uα ×[s,∞)
=
X
α,ī,τ ∈Tε (s)
(5.6)
Z
κε (Θα (x) − Θα (φ−t (x)))JΘα (x)Fα,ī,z (x, t − s).
Ωτ
Note that, for a = <(z), we have
sup |JΘα (x)Fα,ī,z (x, t − s)| ≤
(x,t)∈Ωτ
C# λ(τ )n−1 e−aλ(τ )+C# (λ(τ )+s)
.
(n − 1)!
Since the number of closed orbits with period between t and t + 1 grows at
most exponentially in t (see [27, Section 18.5]),42 we have, for each L > 1,
Z
X
κε (Θα (x) − Θα (φ−t (x)))JΘα (x)Fα,ī,z (x, t − s)
α,ī,τ ∈T (L) Ωτ
(5.7)
Z ∞
n−1 −(a−C# )t+C# s
t e
n −C# Ln+ a2 s
≤ C#
dt
≤ C#
e
(n − 1)!
Ln
42
In fact, it is bounded by C# ehtop (φ1 )t and there is even an asymptotic formula, see
[34], which we are improving in Theorem 2.7
40
5 FLAT TRACES
provided a ≥ C2 for some appropriate constant C2 . From now on we will
consider only closed orbits in Tε (s, L) = {τ ∈ Tε (s) : λ(τ ) ≤ L}. It
. e
.
.
α
is then convenient to pass to charts, (ξ, t) = Θ
α (x, t) = (Θα (x), t), φ−t =
Θα ◦ φ−t ◦ Θ−1
α . Thus, the expression (5.6) can be rewritten as,
Z
X
e −1
(5.8)
κε (ξ − φα−t (ξ))Fα,ī,z ◦ Θ
α (ξ, t − s).
α,ī,τ ∈Tε (s,L)
e α (Ωτ ∩(Uα ×R))
Θ
e α (Ωτ ∩ (Uα × R+ )) consists of ε-neighborhood of a finite
Note that Θ
number of lines (the connected pieces of Θα (τ ∩ Uα )), let us call {Ωα,m } the
collection of such connected components.
˜ ξd ). By construction, for each α, m, there exists a unique
Let ξ = (ξ,
pα,m ∈ Rd−1 such that (pα,m , 0) ∈ Θα (τ ∩ Uα ) and (pα,m , 0, λ(τ )) ∈ Ωα,m .
˜ by [φα
˜ 0)]d = 0 such
Next, define the return time function rα,m (ξ)
(ξ,
−rα,m (ξ̃)
˜ α,m ) = λ(τ ). Note that φα (pα,m , 0) = (pα,m , 0), moreover
that rα,m (ξ)(p
−λ(τ )
α ˜
α
˜
˜ + ξd ).
we have φ (ξ, ξd ) = φ
(ξ, 0) + (0, −t + rα,m (ξ)
−t
−rα,m (ξ̃)
Let us now consider the changes of variables Ξα,m : Ωα,m → Rd+1 , which,
for (ξ, t) ∈ Ωα,m , is given by
.
Ξα,m (ξ, t) = (φα−t (ξ)1 − ξ1 , . . . , φα−t (ξ)d−1 − ξd−1 , ξd , t)
.
.
= (ζ1 , . . . , ζd−1 , ξd , t) = (ζ, ξd , t).
∂(φα )
Let us introduce the matrix Λ of components Λi,j = ∂ξ−tj i − δi,j , i, j ∈
{1, . . . , d − 1}. Note that det(DΞα,m ) = det(Λ) 6= 0 since φα−t is hyperbolic.
Moreover if Ξα,m (ξ, t) = Ξα,m (ξ 0 , t0 ), then t = t0 , ξd = ξd0 . Thus, recalling the
definition of Ωτ , we have kξ − ξ 0 k ≤ C3 ε. Lastly, note that the restriction of
Ξα,m to the first d − 1 coordinates is locally invertible; thus ξ = ξ 0 . Accordingly, Ξα,m is a local diffeomorphism. Hence, there exists εL > 0 such that,
for all ε < εL , Ξα,m is invertible in each Ωτ , τ ∈ Tε (s, L).
Thus Ξα,m (pα,m , 0, t) = (0, 0, t). Hence we have the following expansions43
kξ˜ − pα,m − Λ(pα,m )−1 ζk ≤ C# eC# λ(τ ) kζk2
|rα,m − λ(τ ) − h∇rα,m (pα,m ), Λ(pα,m )−1 ζi| ≤ C# eC# λ(τ ) kζk2 .
By first performing the change of variables Ξα,m , then using the properties
of the chosen κ, and finally using the change of coordinates η = ε−1 ζ, v =
43
Indeed, kφt kC 2 ≤ C# eC# |t| for all t ∈ R.
41
5 FLAT TRACES
ε−1 (t − λ(τ )), we can write (5.8) as
X X Z
−1
−d
−1
e −1
Fα,ī,z ◦ Θ
α ◦ Ξα,m (ζ, ξd , t − s)ε κd−1 (ε ζ)
α,ī,m τ ∈Tε (s,L)
Ξα,m (Ωα,m )
κ1 ε−1 (t − λ(τ ) − h∇rα,m (pα,m ), Λ(pα,m )−1 ζi + O(eC# λ(τ ) ζ 2 ))
×
dζdξd dt
| det(Λ(pα,m )|
Z
X
e −1 ◦ Ξ−1 (0, ξd , λ(τ ) − s)κd−1 (η)| det(Λ)|−1
=
Fα,ī,z ◦ Θ
α
α,m
Rd+1
α,ī,m,τ ∈Tε (s,L)
−1
× κ1 v − h∇rα,m (pα,m ), Λ(pα,m ) ηi dηdξd dv + O
ελ(τ )n eC# λ(τ )
(n − 1)!
.
In the above integral we first integrate first with respect to dv and then with
respect to dη obtaining
Z +δ
X
e −1 (pα,m , ξd , λ(τ ) − s)| det(Λ)|−1 dξd .
Fα,ī,z ◦ Θ
α
α,ī,τ ∈T (s),m
−δ
Define Dhyp φλ(τ ) = Λ(pα,m ) + 1. If we compute this a matrix in a different
chart or at another point in the orbit we simply obtain a matrix conjugated
to Dhyp φλ(τ ) . Thus | det(Λ)| = | det(1 − Dhyp φλ(τ ) )| depends only on τ . In
addition, we have44
X
X
tr(Λ` Dφα−λ(τ ) ) =
det(Dφα−λ(τ ) )k̄,k̄ =
det(Dhyp φ−λ(τ ) )k̄,k̄
k̄∈I`
k̄∈I` (d−1)
(5.9)
`
= tr(Λ (Dhyp φ−λ(τ ) )),
which, again, depends only on the orbit τ . Accordingly,
X
hωα,ī , φ∗−λ(τ ) ωα,ī ipα,m = tr(∧` (Dhyp φ−λ(τ ) ).
ī
Thus, by summing over the connected components of Ωα,m , and taking into
account the multiplicity of orbits, and by resumming on α and recalling (2.4)
44
Given a vector space V d over R and a matrix representation of a linear operator
A : V d → V d , we can construct, by the standard external product, a matrix representation
of Λ` A : Λ` (V d ) → Λ` (V d ) with elements aī,j̄ = det(Aī,j̄ ) (see definition (A.1) or [7]). By
det(A)k̄,k̄ we mean the determinant of the matrix (Aki ,kj ). Note that det(Dφα
−λ(τ ) )k̄,k̄ = 0
unless k` = d. Also, let I` (d − 1) be the set of ordered multiindex i1 < · · · < i` < d.
42
6 TENSORIAL OPERATORS
we finally obtain
Z
n−1
X
X χ` (τ ) [λ(τ ) − s]
λ(τ
)
(`)
n
(`)
hω
,
R
(z)
L
(j
)i
−
x
α,
ī
,α,
ī,x
s
(n − 1)!µ(τ )ez(λ(τ )−s) (5.10)
M α,ī
τ ∈T (s)
a
a
n −C# Ln+ 2 s
≤ C# ε(a − C2 )−n e 2 s + C#
e
,
for each L large enough and ε < εL . Taking the limit ε → 0 followed by
L → ∞ yields the Lemma.
By a direct computation using Lemma 5.1 we have the natural formula45
Z ∞
sn−1
e−zs
tr[ (R(z)Ls ) = tr[ (R(z)n+1 ).
(5.11)
(n
−
1)!
0
6
Tensorial Transfer Operators
In this section we extend the methods of Liverani-Tsujii [32] to the case of
flows. The goal is to provide a setting in which a formula of the type (5.1)
makes sense and can be used to compute the trace (see Lemma 6.8 for the
exact implementation). The first step is to note that, by equation (A.4), the
(`)
adjoint (with respect to h·, ·iΩ` ) of Lt is given by, for g ∈ Ω`0,r ,
(`)
.
Lt g = (−1)`(d−`) ∗(φ∗t (∗g)).
(`)
(6.1)
(`)
Next we would like to take the tensor product of Lt times Lt , and define
2 46
a Banach space, connected to the product space Ω2`
on which it acts
r (M ),
naturally. Note that, contrary to the discrete case, in the continuous setting
this procedure naturally yields a R2+ action in the variables s, t ≥ 0 (rather
than a flow). We start with the construction of the Banach space.
45
Indeed,
Z
0
46
∞
e−zs
X Z λ(τ ) sn−1 χ` (τ )
sn−1
tr[ (R(z)Ls ) =
λ(τ )e−zλ(τ ) .
(n − 1)!
(n − 1)! µ(τ )
0
τ ∈T
By M 2 we mean the Riemannian manifold with the product metric.
43
6.1 Spaces and Operators
6.1
6 TENSORIAL OPERATORS
Spaces and Operators
We use the construction developed in section 3 applied to the manifold M 2 .
First of all consider the atlas {Uα , Θα }α∈A chosen at the beginning of
section 4.47 We define the map I(u, s, t, u0 , s0 , t0 ) = (s, u0 , u, s0 , t, t0 ), u, u0 ∈
Rdu , s, s0 ∈ Rds , t, t0 ∈ R, the atlas {Uα × Uβ , I ◦ (Θα × Θβ )}α,β∈A and the
partition of unity {ψα,β }α,β∈A , where ψα,β (x, y) = ψα (x)ψβ (y). We are thus
in the situation of section 3 with d1 = du + ds = d − 1 and d2 = d + 1. Note
that condition the conditions (3.1) are automatically satisfied.48 We choose
the cones given by the choice ρ = 2, ρ− = 1 and the set of “stable” leaves,
which we denote by Σ2 , as in footnote 21.
Remark 6.1. From now on we will ignore the map I since it is just a trivial
permutation of the coordinates.
2
By appendix A we can consider the exterior forms Ω2`
r (M ) and the re2
lated scalar product. We define the projections πi : M → M , i ∈ {1, 2}, such
that π1 (x, y) = x and π2 (x, y) = y. For each pair of `-forms f, g in Ω`r (M )
2
`
we have that π1∗ f ∧ π2∗ g ∈ Ω2`
r (M ). In addition, given a, b, f, g ∈ Ωr (M ), by
equation (A.1), we have
hπ1∗ a ∧ π2∗ b, π1∗ f ∧ π2∗ gi(x,y) = ha, f ix hb, giy .
(6.2)
Moreover, by (A.2), it follows
∗ (π1∗ f ∧ π2∗ g) = (−1)`(d−`) π1∗ (∗f ) ∧ π2∗ (∗g).
(6.3)
.
Next we define the vector space Ω`2,r (M ) = span{π1∗ f ∧π2∗ g : f, g ∈ Ω`0,r (M )}
on which we intend to base our spaces. Note that locally the C s closure of
Ω`2,s (M ) contains all the forms h such that i(V,0) h = i(0,V ) h = 0. Moreover h
can be written as
X
X
(x, y)ωα,ī (x) ∧ ωβ,j̄ (y),
(6.4)
h=
hα,β =
ψα,β (x, y)hα,β
ī,j̄
α,β
α,β,ī,j̄
.
where hα,β = ψα,β h, and hα,β
∈ C s̄ (M 2 ), for each s̄ > s.49
ī,j̄
47
Note that {Uα , Θα } is naturally adapted to the flow φ−t which has ds dimensional
unstable manifold and du dimensional stable manifold.
48
Note that the third condition holds both for x2d and x2d−1 .
49
This follows since C ∞ is dense in C s̄ in the C s topology and smooth functions can
be approximated by tensor functions, e.g. by Fourier series, in any C s norm. Note that if
s ∈ N, then one can choose s = s̄ (for a more refined description, not used here, see the
theory of little Hölder spaces).
44
6.2 Lasota-Yorke inequalities
6 TENSORIAL OPERATORS
Lastly, we must choose an appropriate set of test functions and vector
fields. For the set of test functions we choose Γ`2,r (Wα,β,G ), defined by the
restriction of span{π1∗ f ∧ π2∗ g : f, g ∈ Ω`r (M ), π1∗ f ∧ π2∗ g|∂Wα,β,G = 0} to
Wα,β,G . The set of vector fields have only the restriction that50
[v, w] = ([(π1 )∗ v, (π1 )∗ w], [(π2 )∗ v, (π2 )∗ w]).
(6.5)
The reader can easily check that the above choices satisfy all the conditions specified in Section 3. Thus we can apply the results in Section 3 and
k·kp,q,`,2
.
call k·kp,q,`,2 the resulting norms and B2p,q,` = Ω`2,r (M )
the corresponding Banach spaces.
(`)
Finally, we define the required operators Lt,s . For each f, g ∈ Ω`r (M )
(`)
.
(`)
(`)
Lt,s (π1∗ f ∧ π2∗ g) = (π1∗ (Lt f )) ∧ (π2∗ (Ls g))
(6.6)
(`)
which extends by linearity to an operator Lt,s : Ω`2,r (M ) → Ω`2,r (M ).
To avoid the problem of small times, for which the cone contraction might
fail, we apply the same strategy used in the previous section: we define
|||h|||p,q,`,2 = sup kLt,s hkp,q,`,2 ,
t,s≤t0
|||·|||p,q,`,2
.
⊂ B2p,q,` . Note that, by construcand we define the spaces Be2p,q,` = Ω`2,r
tion, we trivially have a set of inequalities as in (3.13).
Following Remark 3.3, equation (A.4), setting ω2 = π1∗ ω̃ ∧ π2∗ ω̃, yields
(d)
Lt,t (f ω2 ) = f ◦ (φ−t × φt )Jφ−t ω2 .
That is, we recover the same type of operators studied in [32].
6.2
Lasota-Yorke inequalities
We can now obtain several results parallel to those in sections 4.1, 4.2. As
the proofs are almost identical to those in section 4.3 we will not give full
details and highlight only the changes that need to be made.
50
In the right hand side the commutators are computed keeping fixed the y coordinate
in the first component of the vector, and keeping fixed x in the second component.
45
6.2 Lasota-Yorke inequalities
6 TENSORIAL OPERATORS
(`)
Lemma 6.2. For each p + q < r − 1, t, s ∈ R+ , Lt,s ∈ L(Be2p,q,` , Be2p,q,` ). More
(`)
precisely Lt,s is an R2+ action over Bep,q,` . Moreover, Lt,0 and L0,s are strongly
continuous semigroups with generators X1 , X2 , respectively. In addition, we
have51
(`) L
h
≤Cp,q eσ` (t+s) |||h|||p,q,`,2
t,s p,q,`,2
(`) ≤Cp,q eσ` (t+s) e−σp,q min{t,s} |||h|||p,q,`,2 + Cp,q eσ` (t+s) |||h|||p−1,q+1,`,2
Lt,s h
p,q,`,2
+
2
X
eσ` (t+s) |||Xj h|||p−1,q+1,`,2 .
j=1
Proof. As in Section 4, we first prove an analogue of Lemma 4.3, and then
we prove a stronger version of it, as in Lemma 4.5.
The proof contained in Subsection 4.3 can be followed almost verbatim.
By the same construction one obtains the equivalent of (4.10), that is
Z
X X Z
(`)
hg, Lt,s hiωvol =
(−1)(d−`)` ψβ,β 0 Jφ−1
t−s
Wα,α0 ,G
β,β 0 ∈A ī,j̄,ī,j̄ 0
e 0
k∈K
Wβ,β 0 ,G
k
β,β
× JW (φt × φ−s )gī0 ,j̄ 0 ◦ (φt × φ−s ) · hωβ,ī , ∗φ∗t ∗ ωα,j̄ ihωβ,ī0 , φ∗−s ωα,j̄ 0 iωvol .
For the case ℘ = 0, the argument is exactly the same as in Section 4.3,
apart from the estimate of the volume of φ−t × φs (Wα,α0 ,G ) to which Appendix C cannot be applied directly. To estimate such a volume, after set.
s
u
ting (z1 , z2 ) = (Θα × Θα0 )−1 (G(0)), let us consider W (z) = W6δ
(z1 ) × W6δ
(z2 )
and the holonomy from Wα,α0 ,G to W (z) determined by the weak unstable
foliation in the first coordinate and the weak stable in the second. Clearly
the distance between the corresponding points in the images of Wα,α0 ,G and
W (z) is uniformly bounded; hence the required volume is proportional to the
volume of φ−t × φs (W (z)), which is bounced by ehtop (φ1 )(t+s) .
For the case ℘ > 0, by equations (6.5), we can reorder the vector fields
as to have first the vector fields tangent to Wβ,β 0 ,Gk , then the vector fields in
the unstable direction of φt × φ−s and then the two neutral directions. We
can then proceed as in equation (4.20). The rest of the argument is the same
as before apart from the fact that the weakest contraction is now given by
51
Here and in the sequel we suppress the λ dependence in the constants to simplify the
notation.
46
6.2 Lasota-Yorke inequalities
6 TENSORIAL OPERATORS
the case in which all the vector fields act on the component with the smallest
time, hence the min{t, s} factor. Then we can proceed exactly as in section
4.
Lemma 6.3. For each p + q < r − 1, ` ∈ {0, . . . , d − 1}, and for each
.
<(z) = a > σ` , the operator
Z ∞Z ∞
1
(`)
(`)
n
(ts)n−1 e−z(t+s) Lt,s dtds
(6.7)
R2 (z) =
2
(n − 1)! 0
0
satisfies
(`) n R2 (z) ≤ Cp,q (a − σ` )−2n
|||h|||p,q,`,2
(|z| + 1)
(`) n +
|||h|||p−1,q+1,`,2 .
≤ Cp,q,a
R2 (z) h
(a − σ` + σp,q
)2n (a − σ` )2n
p,q,`,2
2
p,q,`,2
Hence R(`) (z) is a linear operator on Be2p,q,` with spectral radius bounded by
(<(z) − σ` )−2 and essential spectral radius bounded by (<(z) − σ` + σp,q
)−2 .
2
Proof. Again the proof follows closely Subsection 4.2, and more precisely
Lemma 4.6. The only difference rests in the need to decompose the domain
of integration of (6.7) into four pieces: A1 = {(t, s) ∈ R2+ : t ≤ t0 , s ≤ t0 },
A2 = {(t, s) ∈ R2+ : t ≤ t0 , s > t0 }, A3 = {(t, s) ∈ R2+ : t > t0 , s ≤ t0 } and
A4 = {(t, s) ∈ R2+ : t > t0 , s > t0 }.
The estimate of the integration over A4 follows verbatim the argument in
Lemma 4.6, except that one then obtains (instead of equation (4.29))
Z ∞Z ∞
Cp,q (ts)n−1
|||h|||p,q,`,2 dtds
2 [a−σ` ](t+s)+min{t,s}σp,q
t0 t0 (n − 1)! e
(6.8)
Cp,q (|z| + 1)
+
|||h|||p−1,q+1,`,2 .
(a − σ` )2n
47
6.3 Trace representation
6 TENSORIAL OPERATORS
Next we estimate the following integral.
Z ∞Z ∞ −[a−σ` ](t+s)−min{t,s}σp,q
Z ∞ Z ∞ −[a−σ` ](t+s)−tσp,q
(ts)n−1
(ts)n−1
e
e
=
2
dt
ds
(n − 1)!2
(n − 1)!2
t0 t0
t0
t
Z ∞
n−1
X
2
tn+k−1 e−(2a−2σ` +σp,q )t
≤
dt
(n − 1)!k! 0
(a − σ` )n−k
k=0
n−1 X
n+k−1
1
=2
n−k
(a − σ` ) (2a − 2σ` + σp,q )n+k
n−1
k=0
n−1 X
n+k−1
(a − σ` )k
−n
−n
≤ 2(2a − 2σ` + σp,q ) (a − σ` )
(2a − 2σ` + σp,q )k
n−1
k=0
σp,q −n
) (a − σ` )−n .
≤ (a − σ` +
2
Similarly, the integrals over A2 , A3 are bounded by
Z ∞ Z t0
e−[a−σ` ](t+s)−sσp,q (ts)n−1
Cp,q t0 n
Cp,q
ds
dt
2
≤
≤
,
2
n
(n − 1)!
(a − σ` ) n!
(a − σ` + σp,q
)2n
0
0
2
(a−σ +
σp,q 2
) t
e
0
`
2
, by analogy with (4.28). The estimate of the
provided n ≥
(a−σ` )
integral over A1 is treated similarly.
6.3
Tensor representation of the trace
Following the scheme of [32], we define a suitable delta like functional acting
on Be2p,q,` and we construct an approximation scheme for such a functional.
For each f, g ∈ Ω`r (M ), we define
Z
.
`
∗
∗
δ2 (π1 f ∧ π2 g) =
hf, gix ωM (x) .
(6.9)
M
Such a definition extends by linearity and density to all sections in the closure
of Ω`2,r (M ) with respect to the C r topology. Thus we obtain
XXZ
`
δ2 (h) =
ψα,β (x, x)hα,β
(x, x)hωα,ī (x), ωβ,j̄ (x)iωM (x)
ī,j̄
α,β
=
ī,j̄
XZ
α,ī
M
M
(6.10)
ψα (x)hα,α
(x, x)ωM (x).
ī,ī
48
6.3 Trace representation
6 TENSORIAL OPERATORS
Given κε as in Definition 3.6, we set J (x, y) = 0 if d(x, y) > δ, otherwise we
set
. X
J (x, y) =
ψα (y) · JΘα (y) · κε (Θα (x) − Θα (y))ωα,ī (x) ∧ ωα,ī (y). (6.11)
α,ī
Lemma 6.4. For each h ∈ Ω`2,s (M ), we have that
limhJ , hiΩ(`) = δ2` (h).
→0
(6.12)
2
Proof. For f, g ∈ Ω`r (M ), let h = π1∗ f ∧ π2∗ g, then
Z
hJ , π1∗ f ∧ π2∗ gi(x,y) ωM (x) ∧ ωM (y)
limhJ , hiΩ(`) =
2,s
→0
2
Z X M
ψα (y)JΘα (y) · κε (Θα (x) − Θα (y))hf, gix ωM (x) ∧ ωM (y).
= lim
→0
M2
α
The result follows, for h of the above form, by integrating with respect to y.
We can extend this result to a generic h by linearity of the scalar product
and by density (see footnote 49).
Next we need the equivalent of Lemma 3.4, the proof is omitted since it
is exactly the same as before, starting from the space Ω`2,r (M ) and choosing
k appropriately.
0
Lemma 6.5. There exists an injective immersion 2 : Be2p,q,` → Ω`2,r (M ) .
0
Lemma 6.6. The current δ2` extends uniquely to an element of Be2p,q,` .
Proof. Since Be2p,q,` is defined by the closure of the sections in Ω`2,r , it suffices
to prove that there exists c > 0 such that δ2` (h) ≤ c |||h|||p,q,`,2 .
.
Let WD = {(x, y) ∈ M 2 : x = y} and recall that δ2` corresponds to
integrating on such a manifold by (6.10). If x ∈ Uα we can foliate WD , in
e given by the graph of the
the local chart Vα , with the manifolds Wα,Gs ∈ Σ
s u
s u
s u
functions Gs (x , y ) = (x , y , s, x , y , s). Accordingly,
Z
XZ
`
δ2 (h) ≤
ds hψα (x)ωα,ī ∧ ωα,ī , hi ≤ C# khk0,q,`,2 .
Wα,Gs
α,ī
49
6.3 Trace representation
6 TENSORIAL OPERATORS
At this point, we would like to make sense of the limit of Jε in Be2p,q,` .
Unfortunately, this can be done only at a price.
(`)
Lemma 6.7. For <(z) sufficiently large, the sequence R2 (z)Lt0 ,t0 J is a
Cauchy sequence in Be2p,q,` . We call δ̄2` (z) the limit of such a sequence.
Proof. Let us start by showing that the sequence is bounded in Be20,q,` . Let
g ∈ Γ`2,r , then by recalling the definition of the norms (see 3.10), of the
(`)
operators R2 (z) and by using equation (A.4)52 we have that
Z
Z
Z
(`)
−z(t+s)
hg, R2 (z)Lt0 ,t0 J i =
ds dt e
Jφ−t−t0 (x)
Wα,β,G
R2+
Wα,β,G
(6.13)
∗
∗
× Jφs+t0 (y)h(∗φt+t0∗ ×φ−s−t0 )g, J )i ◦ φ−t−t0 (x) × φs+t0 (y).
Note that, provided t0 has been chosen large enough, the tangent spaces of
the manifold φ−t−t0 × φs+t0 (Wα,β,G ) are strictly contained in the cone C 1 .
2
∞
Let
p
e
∈
C
(R,
R
),
supp(e
p
)
⊂
(−δ,
δ),
p
e
(−t)
=
p
e
(t),
be
such
that
+
P
p
e
(t
+
nδ)
=
1
for
all
t
∈
R.
Let
n∈Z
.
Fn,m (x, y, t, s) = Jφ−t−t0 (φnδ+t0 (x))Jφs+t0 (φ−mδ−t0 (y))
−1
× det D (φ−nδ−t0 × φmδ+t0 )|T Wα,β,G
◦ φnδ+t0 × φ−mδ−t0 (x, y).
.
By the change of variable (x0 , y 0 ) = φ−nδ−t0 × φmδ+t0 (x, y), (6.13) becomes
X Z
ds dt pe(−t + nδ)e
p(s − mδ)e−z(t+s)
n,m∈N
Z
×
R2+
Fn,m · h(∗φ∗t+t0 ∗ ×φ∗−s−t0 )g, J i ◦ φ−t+nδ × φs−mδ .
φ−nδ−t0 ×φmδ+t0 (Wα,β,G )
Note that if (x, y) ∈ φ−nδ−t0 × φmδ+t0 (Wα,β,G ) and y ∈ supp ψγ , then the
integrand is different from zero only if x ∈ Uγ . Thus, for each connected
c of φ−nδ−t0 × φmδ+t0 (Wα,β,G ) ∩ supp ψβ × supp ψγ , by the same
component W
arguments used in the proof of Lemma 3.7, there exists a manifold Wγ ∈ Σ2 ,
Wγ ⊂ Uγ,γ such that ∂Wγ ∩ [supp ψγ ]2 ⊂ ∂([supp ψγ ]2 ) and the property
c equals the integral on Wγ for each f .53
that the integral of hf, Jε i on W
52
To apply such a formula to the present case, we have to use it together with equations
(6.2), (6.3), and then extend it by density.
53
Since Jε is different from zero only if both coordinates belong to Uγ .
50
6.3 Trace representation
6 TENSORIAL OPERATORS
Let {Wγ,k }Kγ,n,m be the collection of such manifolds. Recalling equation
(6.11), we can then rewrite the previous formula54 by setting t0 = +t − nδ,
s0 = s − mδ. Thus we obtain
X X
X Z
0
0
ds dt pe(t0 )e
p(s0 )e−z(t +s +(n+m)δ)
n,m∈N γ∈A,ī∈I k∈Kγ,n,m
Z
R2
Fn,m (x, y, t0 + nδ, s0 + mδ) · ψγ ◦ φs0 (y)JΘγ ◦ φs0 (y)
×
Wγ,k
× κε (Θγ (φ−t0 (x)) − Θγ (φs0 (y)))
× h(∗φ∗t0 +t0 +nδ ∗ ×φ∗−s0 −t0 −mδ )g, ωγ,ī ∧ ωγ,ī i ◦ φ−t0 × φs0 .
Recall that the manifolds Θγ (Wγ,k ) are graphs of the type55
.
Gk (xs , y u ) = (xs + x̂s , H u (xs , y u ), H s (xs , y u ), y u + ŷ u , H10 (xs , y u ), H20 (xs , y u ))
u
s
u
s
where, since T Θ−1
γ (Wγ,k ) ∈ C 21 , max{k∂x H k + k∂x H k, ∂y H k + k∂y H k} ≤
1
. It is then natural to set Gk = Θ−1
γ ◦ Gk and
2
s u
Fbγ,n,m,k,ī (xs , y u , s, t) = Fγ,n,m (Θ−1
γ ◦ Gk (x , y ), t + nδ, s + mδ)
s u
s u
× [JΘγ · ψγ ] ◦ Θ−1
γ (Gk (x , y ) + (0, 0, 0, s)) × JGk (x , y )
s u
× h(∗φ∗t+t0 +nδ ∗ ×φ∗−s−t0 −mδ )g, ωγ,ī ∧ ωγ,ī i ◦ Θ−1
γ (Gk (x , y ) + (0, 0, t, s)).
We can then continue our computation and write
X X
X Z
ds dt pe(t)e
p(s)e−z(t+s+(n+m)δ)
n,m∈N γ∈A,ī∈I k∈Kγ,n,m
Z
×
R2
dxs dy u Fbγ,n,m,k,ī (xs , y u , t, s)
(6.14)
Rd−1
−d
× ε κd−1 (ε−1 (xs − H s (xs , y u ) + x̂s , H u (xs , y u ) − y u − ŷ u ))
× κ1 (ε−1 (H 0,1 (xs , y u ) − H 0,2 (xs , y u ) − t − s)).
54
To be precise we should treat separately the terms with n = 0 or m = 0, we leave this
as an exercise for the reader since it is quite simple to handle, although a bit cumbersome.
55
We drop the subscript n, m, γ, k from x̂, ŷ, H u , H s , H 0,1 , H 0,2 in the following equation
since it is clear that such graphs depends from all these choices. Moreover we drop n, m, γ
from Gk since it will be always be clear which open set we are considering.
51
6.3 Trace representation
6 TENSORIAL OPERATORS
Note that we havekFbkC 0 ≤ C# kgkC 0 as in equation (4.14). In addition, the
map Ξ defined by
ξ = xs − H s (xs , y u ) + x̂s ,
η = H u (xs , y u ) − y u − ŷ u ,
t0 = H 0,1 (xs , y u ) − H 0,2 (xs , y u ) − t − s,
s0 = s
(6.15)
is locally invertible, hence it can be used as a change of variables. Thus,
setting a = <(z), we can bound (6.14) by
X X X
X
Cz e−a(n+m)δ kgkC 0 ≤ Cz
e(htop (φ1 )−a)(n+m)δ kgkC 0
n,m∈N γ∈A k∈Kγ,n,m
n,m∈N
≤ Cz (a − htop (φ1 ))−2 kgkC 0 .
Which, taking the sup on the manifolds and test forms, yields
(`)
(`)
≤ Cz (a − htop (φ1 ))−2 .
R2 (z)Lt0 ,t0 Jε 0,q,`,2,2
Next, given ε > ε0 > 0, by using equation (6.14), (6.15) and the intermediate
value theorem we have, for <(z) large enough,
Z
(`)
(`)
0
hg, R2 (z)Lt0 ,t0 J − R2 (z)Lt0 ,t0 J i
Wα,β,G
Z
X X
X
0
dξ dη dt0 ds0 e−zs
≤ C#
n,m∈N γ∈A,ī∈I k∈Kγ,n,m
Rd+1
× Fγ,n,m,k,ī (ξ, η, t0 , s0 ) − Fγ,n,m,k,ī (0, 0, 0, s0 ) · [κε ((ξ, η, t0 )) − κε0 ((ξ, η, t0 ))]
≤ Cz εmin{q,1} kgkC q
where Fγ,n,m,k,ī are functions which satisfy kFγ,n,m,k,ī kC p ≤ C# eC# (n+m) . This
proves the Lemma for Be20,q,` . The extension to Be2p,q,` is treated similarly after
integrating by parts p times.
We can finally present a description of the trace that does not involve any
limit (although, unfortunately, not for the operators we are interested in).
Lemma 6.8. For each z ∈ C, <(z) large enough, n ∈ N and s, t ∈ R+ , we
have that
(`)
(`)
(`) (`)
(`)
δ2 R2 (z)n Lt,s δ̄2 = tr[ (R(`) (z)2n Lt+s+2t0 ).
(6.16)
52
6.3 Trace representation
6 TENSORIAL OPERATORS
. (`)
(`)
(`) (`)
Proof. Let Rn (z, s, t) = δ2 R2 (z)n Lt,s δ̄2 . From Lemma 6.7 and equation (6.10) we obtain
α,α
XZ
(`)
(`)
Rn (z, s, t) = lim
ωM (x) ψα (x) R2 (z)n Lt+t0 ,s+t0 J
(x, x)
→0
= lim
α,β,ī,j̄
0
0
(t0 s0 )n−1 e−z(t +s )
ds dt
[(n − 1)!]2
R2+
X Z
→0
ī,ī
M
α,ī
0
0
Z
ωM (x)ψα (x) [ψβ JΘβ ] ◦ φs0 +s+t0 (x)
M
× κ (Θβ (φ−t0 −t−t0 (x)) − Θβ (φs0 +s+t0 (x)))
× (−1)`(d−`) hφ∗−t0 −t−t0 ωβ,j̄ , ωα,ī ix hωα,ī , ∗φ∗s0 +s+t0 ∗ ωβ,j̄ ix .
Next, we sum over ī, α and use (A.4) to obtain
0 0 n−1 −z(t0 +s0 ) Z
XZ
e
0 0 (t s )
ds dt
R(z, s, t) = lim
[ψβ JΘβ ] ◦ φs0 +s+t0 (x)
→0
[(n − 1)!]2
R2+
M
β,j̄
× κ (Θβ (φs0 +s+t0 (x)) − Θβ (φ−t0 −t−t0 (x)))
× Jφs0 +s+t0 hφ∗−t0 −t−s0 −s−2t0 ωβ,j̄ , ωβ,j̄ i ◦ φs0 +s+t0 (x)
0 0 n−1 −z(t0 +s0 ) Z
XZ
e
(`)
0 0 (t s )
hωβ,j̄ , Lt0 +s0 +t+s+2t0 j,β,j̄,x i,
= lim
ds dt
2
→0
2
[(n
−
1)!]
M
R+
β,j̄
where, in the last line, we have changed variables and used (5.2). Next, after
the change of variables v = t0 + s0 , u = t0 , we integrate in u (the integral
is given by the β-function) and recall (5.3) to obtain the statement of the
Lemma.
At last we can start harvesting the benefits of the previous results. By
(`)
(`)
(`)
(`)
Lemma 6.3 we have R2 (z) = P2 (z) + U2 (z) where P2 (z) is a finite rank
(`)
operator and the spectral radius of U2 (z) is bounded by (<(z)−σ` + σp,q
)−2 .
2
Recall that by Lemma 4.6 we have R(`) (z) = P (`) (z) + U (`) (z) where P (`) (z)
is a finite rank operator and the spectral radius of U (`) (z) is bounded by
(<(z) − σ` + σp,q )−1 . In addition,
P (`) (z)U (`) (z) = U (`) (z)P (`) (z) = 0 ;
(`)
(`)
(`)
(`)
P2 (z)U2 (z) = U2 (z)P2 (z) = 0.
Lemma 6.9. There exists C4 > 0 such that, for each t, s ∈ R+ , n ∈ N and
z ∈ C, a = <(z) ≥ C4 , we have that
(`)
(`)
(`)
δ2 P2 (z)n Lt,s δ̄2 (z) = tr P (`) (z)2n Lt+s+2t0 .
(6.17)
53
6.3 Trace representation
6 TENSORIAL OPERATORS
Proof. Recalling (6.6) and (A.4) we can write, for <(z) sufficiently large,
(`)
hR2 (z)n Lt+t0 ,s+t0 J , π1∗ (f ) ∧ π2∗ (g)iΩ`2,s
Z
(6.18)
(t0 s0 )n−1
∗
∗
∗
∗
= ds0 dt0
hJ
,
π
(∗φ
∗f
)
∧
π
(φ
g)i.
0
0
1
t +t+t0
2
−s−s −t0
2 z(t0 +s0 )
R2+ ×M 2 (n − 1)! e
By taking the limit → 0, since the integrals are uniformly convergent
with respect to time (as in equation (5.6), by a rough bound on the growth
of the number of orbits), and recalling Lemma 6.7 and equation (6.12) we
obtain
Z 0 0 n−1
(t s ) h∗φ∗t0 +t+t0 ∗ f, φ∗−s0 −s−t0 gi
(`)
n
`
∗
∗
R2 (z) Lt,s δ̄2 (π1 (f ) ∧ π2 (g)) =
(n − 1)!2 ez(t0 +s0 )
R2+ ×M
Z
0
0 Z
(t0 s0 )n−1 e−z(t +s )
`(d−`)
hf, φ∗−s−t−s0 −t0 −2t0 gi
= (−1)
ds dt
2
2
(n
−
1)!
M
R+
Z 1
Z
Z
2n−1 n−1
(6.19)
τ
η (1 − η)n−1
`(d−`)
∗
hf,
φ
gi
dη
dτ
= (−1)
−τ
−t−s−2t
0
(n − 1)!2 ezτ
M
0
R+
Z
2n−1
∗
τ
hf, φ−τ −t−s−2t0 gi
= (−1)`(d−`)
dτ
(2n − 1)!ezτ
R+ ×M
= (−1)`(d−`) hf, R(z)2n Lt+s+2t0 giΩ` ,
since the dη integral in the third line is given by the β-function. By the von
Neumann expansion, for <(z) and <(ξ) large enough, we have
(`)
(ξ1−R2 (z))−1 Lt,s δ̄2` (π1∗ (f )∧π2∗ (g)) = (−1)`(d−`) hf, (ξ1−(R(z)(`) )2 )−1 Lt+s+2t0 giΩ` .
)−2 },
Since both expression are meromorphic in the region {|ξ| > (a−σ` + σp,q
2
it follows that they must agree on such a region. Given a curve γ surrounding
the region {|ξ| ≤ (a − σ` + σp,q
)−2 }, we can use standard functional analytic
2
56
calculus to obtain
Z
−1
h
im
1
(`)
(`)
`
Lt,s δ̄2` (h)dξ
U2 (z) Lt,s δ̄2 (h) =
ξ m ξ1 − R2 (z)
2πi γ
Z
−1
(−1)`(d−`)
=
ξ m hf, ξ1 − R(`) (z)2
Lt+s+2t0 giΩ`s dξ
2πi
γ
= hf, U (`) (z)2m Lt+s+2t0 giΩ`s .
56
See [25], for example.
54
6.3 Trace representation
6 TENSORIAL OPERATORS
Furthermore, we have
h
im
(`)
P2 (z) Lt,s δ̄2` (h) = hf, P (`) (z)2m Lt+s+2t0 giΩ`s .
(6.20)
Since P (`) (z)2m is a finite rank operator, it follows that P (`) (z)2m Lt+s+2t0
0
is also finite rank on Bp,q,` . Thus there exist um,k ∈ Bp,q,`
, vm,k ∈ Bp,q,` ,
k ∈ {1, . . . , L}, such that
hf, P
(`)
(z)
2m
Lt+s+2t0 giΩ`s =
L
X
hf, vm,k iΩ`s um,k (g).
k=1
Next, we define the mollification of an element h ∈ Ω`2,s by
XX
M2,ε (h)(x, y) =
ψα (x, y)ψβ (x, y)ωα,ī (x) ∧ ωβ,j̄ (y)
α,β
×ε
−2d
ī,j̄
(6.21)
Z
κε (Θα (x) − ξ)κε (Θβ (y) −
R2d
−1
η)hα,β
(Θ−1
α (ξ), Θβ (η)).
ī,j̄
By duality, we can define the mollificator M02,ε on the currents. Following
the same reasoning as in Lemma 3.7 we have that M02,ε , restricted to B2p,q,` ,
is a bounded operator which converges, in the sense of Lemma 3.7, to the
identity when ε → 0.
A direct computation shows that M2,ε (π1∗ (f )∧π2∗ (g)) = π1∗ (Mε f )∧π2∗ (Mε g),
hence (6.20) implies
im
nh
o
(`)
0
`
M2,ε P2 (z) Lt,s δ̄2 (h) = hMε f, P (`) (z)2m Lt+s+2t0 Mε giΩ`s .
Note that the value on the product sections determine uniquely
the current
PL
(`)
∗
0
m
`
(see footnote 49). In fact, since M2,ε [P2 (z)] Lt,s δ̄2 and k+1 π1 (M0ε vm,k ) ∧
π2∗ (M0ε um,k ) agree on each element of the type π1∗ (f )∧π2∗ (g), they agree as currents. Thus, by Lemma 6.5, they are the same element of B2p,q,` . Accordingly,
by Lemma 6.6 and equation (6.9), we finally obtain that
(`)
(`)
δ2 P2 (z)m Lt,s δ̄2` = lim δ2 M02,ε P2 (z)m Lt,s δ̄2`
ε→0
= lim
ε→0
= lim
ε→0
L
X
δ2 (π1∗ (M0ε vm,k ) ∧ π2∗ (M0ε um,k ))
k=1
L
X
hM0ε vm,k , M0ε um,k i
k=1
=
L
X
k=1
55
um,k (vm,k ),
6.3 Trace representation
6 TENSORIAL OPERATORS
Thus we reduced the computation of the “flat trace” to that of a trace of
a matrix.
Lemma 6.10. For each p + q < r − 1, z = a + ib ∈ C with a > C4 ,
)−1 , we have, for each n ∈ N, that
µ > (a − σ` + σp,q
2
[ (`) n
tr (R (z) ) − tr(P (`) (z)n ) ≤ Cp,q,a,µ µn .
(6.22)
Proof. Lemma 6.8 yields, for all n, m, k ∈ N, |m − k| ≤ C# , the following
ugly, but useful, formula
Z ∞ Z ∞ m−1 k−1 −z(t+s)
s
t e
(`)
(`)
(`)
n (`)
dt ds
δ
R2 (z) Lt−t0 ,s−t0 δ̄2
(m − 1)!(k − 1)! 2
t0
t0
Z ∞ Z t−t0 m−1
s
(t − s)k−1 e−zt [ (`) 2n
=
dt
tr (R (z) Lt )
ds
(m − 1)!(k − 1)!
2t0
t0
(6.23)
Z 2t0
tm+k−1
[
(`)
2n+m+k
−zt [
(`)
2n
= tr (R (z)
)−
dt
e tr (R (z) Lt )
(m + k − 1)!
0
Z ∞ Z t0 m−1
s
(t − s)k−1 e−zt [ (`) 2n
tr (R (z) Lt ).
−2
dt ds
(m − 1)!(k − 1)!
2t0
0
On the other hand by the spectral decomposition of R2 , as in Lemma 6.3,
and Lemma 6.9 follows, for each µ > µ1 > (a − σ` + σp,q
)−1 ,
2
∞
∞
sm−1 tk−1 e−z(t+s) (`) (`) n (`)
(`)
R2 (z) Lt−t0 ,s−t0 δ̄2
δ
dt ds
(m − 1)!(k − 1)! 2
t0
t0
Z ∞ Z ∞ m−1 k−1 −z(t+s)
s
t e
Cp,q,µ1 µ2n
(`)
(`)
(`)
1
n (`)
= dt ds
δ
P2 (z) Lt−t0 ,s−t0 δ̄2 + O
(m − 1)!(k − 1)! 2
(a − σ` )m+k
t0
t0
Z ∞ Z ∞ m−1 k−1 −z(t+s) s
t e
Cp,q,µ1 µ2n
1
(`)
2n (`)
= dt ds
tr P (z) Lt+s + O
(m − 1)!(k − 1)!
(a − σ` )m+k
t0
t0
Z
Z
Finally, recall again that the number of periodic orbits of length t grows at
most exponentially. Hence, by Lemma 5.1, there
exists A ≥ σ` such that
(`)
[
2n
−2n At
| tr (R(z) Lt )| ≤ C# (a − A) e while | tr P (`) (z)2n Lt+s | ≤ C# (a −
σ` )−2n eσ` t . We can use such estimates to evaluate the integrals in (6.23) and
56
7 CONTACT FLOWS
obtain,
[ (`) 2n+m+k
C# t 0 m
Cp,q,µ1 µ2n
1
tr (R (z)
) − tr P (`) (z)2n+m+k ≤
+
(a − σ` )m+k (a − A)2n+k m!
C# (2t0 )m+k
+
.
(a − A)2n (m + k)!
To conclude we choose m = ζn, |m − k| ≤ 1. Note that the last two terms
C# et0
are smaller than µ2n+m+k provided n ≥
, while the first term on
1+ 2 2+ 2
ζ(a−A)
ζ
µ
ζ
the left is bounded by C# µ2n+m+k provided we choose ζ =
7
ln µµ−1
1
−1
µ (a−σ`−1 )
.
Contact flows
The results of the previous sections suffice to prove that the ζRuelle is meromorphic, yet provide very little information on the location of its zeroes and
poles. Such a knowledge is fundamental to extract information from the ζ
functions (e.g. counting results). In section 8 we provide an approach to
gain such informations, partially inspired by [31], based on a Dolgopyat type
estimate on the norm of the resolvent. We are not aware of a general approach to gain such estimates apart for the case of C 1 or Lipschitz foliations
[14, 44, 45] and the case of contact flows ([29, 46]). In the following we will
restrict to the latter since it covers the geometrically relevant case of geodesic
flows in negative curvature. Our approach follows roughly [29, Section 6] but
employs several simplifying ideas, some from [5], some new.
In the case of contact flows d must be odd, du = ds and we can, and
will, assume that the contact form α in coordinates reads (see [5, section 3.2,
Appendix A] for details)
∗
s
u .
(Θ−1
β ) α = dxd − hx , dx i = α0
where (xs , xu , xd ) ∈ Rds × Rds × R = Rd is a point in the chart.
The extra information that we need, and that can be gained in the contact
flow case, are bounds on the size of the resolvent Rds (z) for =(z) large.
Remark 7.1. Since in the following we are mainly interested only in the
case ` = ds we will often drop the scripts ds in the relevant objects.
57
7.1 Dolgopyat’s estimate
7.1
7 CONTACT FLOWS
Dolgopyat’s estimate
Let a = <(z) > σds be fixed once and for all. For a fixed C5 > 0, to be
.
chosen later small enough, define ca = C5 (a − σds )−1 e−1 and
Z ∞
tn−1
b
Lt h.
dt e−zt
Rn (z)h =
(n − 1)!
ca n
then, bounding e−σds t |||Lt |||0,q,ds by one, integrating and using Stirling formula, for each q > 0 gives that
bn (z))h
≤ C# (a − σds )−n C5n |||h|||0,q,ds .
(7.1)
(R(z)n − R
0,q,ds
b
(z)
Moreover, for ca n > t0 , it is easy to check that R
n
0,q,ds
b
≤ C# R
(z)
n
0,q,ds
Thus, it suffices to estimate the latter norm.
In the following arguments it turns out to be convenient to introduce
norms similar to the one used in [29]. We did not use them in the previous
sections since they do not allow to keep track of the higher regularity of the
flow.
Definition 7.2. We define norms k·ksq as the norm k·k0,q,ds with the only
difference that the set Σ is replaced by the set Σs , defined as the elements of
Σ which are subsets of a strong stable manifold. Also, let V u be the set of
continuous vector fields tangent to the strong unstable direction, of norm one,
.
and C 1 when restricted to any unstable manifold.57 We then define khku =
∗ .
s
u
supv∈V u kLv hk0,0,ds . To conclude we define the norm k·kq = k·kq + k·k .
Before starting the real work let us fix some notation and recall some
facts.
Remark 7.3. Let $ be the Hölder regularity of the strong foliations and $∗
the Hölder regularity of the Jacobian of the associated holonomy. Then, in
the contact flow case, $ ≥ λ2λ+ and $∗ ≥ λλ+ .58 For a manifold W ∈ Σs
f = ∪|t|≤δ φt W . For each two nearby manifolds W, W 0 ∈ Σ, let HW,W 0 :
let W
f→W
f 0 be the unstable holonomy, then kJHW,W 0 kC $∗ + kHW,W 0 kC $ ≤ C# .
W
We set $̂ =
2λ
,
λ+
$0 = min{1, $̂}.
57
Note that such vector fields determine a unique flow on each unstable manifold, hence
a unique global flow, even if they are only continuous on the manifold. Thus the corresponding Lie derivative is well defined.
58
See [30] for a quick explanation and references or [21, 22, 23] for more details.
58
.
7.1 Dolgopyat’s estimate
7 CONTACT FLOWS
f ) and
Lemma 7.4. For each q > 0, α ∈ A, W, W 0 ∈ Σα , g ∈ Γd0s ,q (W
ds
h ∈ Ωr , let ḡ = g ◦ HW,W 0 JHW,W 0 . We have that
Z
Z
≤ C# d(W, W 0 ) khku .
hg,
hi
−
hḡ,
hi
f0
W
f
W
fτ }τ ∈[0,1] be a smooth foliation interpolating between W
f and
Proof. Let {W
d
f 0 . Also let Hτ be the holonomy between W
f and W
fτ . Define gτ ∈ Γ0s ,0 (W
fτ )
W
P
.
−1
−1
by gτ (x) = ī (gī · JHτ ) ◦ Hτ (x) ωα,ī (x), then
Z
Z
hg1 , hi −
f0
W
1
Z
hg0 , hi =
f
W
0
d
dτ
Z
Z
hgτ , hi =
fτ
W
0
1
d
dτ
Z X
f
W
gī hī ◦ Hτ .
ī
By the implicit function theorem it follows that dτd hī ◦ Hτ = (Lv hī ) ◦ Hτ for
some v ∈ V u . From this the Lemma readily follows.
Lemma 7.5. For each η ∈ (0, $∗ ), t ∈ (t0 , ∞) and h ∈ Ωdr s we have
khk0,η,ds ≥ khksη ,
khk0,1+η,ds ≤ C# khk∗η .
Proof. The first inequality is obvious since the sup in the definition of the
norm is taken on a larger set. To prove the second note that Lemma 7.4 allows
us to compare the integral on a manifold in Σ with a close by manifold in Σs
implying
khk0,1+η,ds ≤ C# khku + C# khksη .
We are now ready to state and prove the main estimate of this section.
Proposition 7.6. Let φt be a contact flow such that $0 > 23 . Then for each
s
η ∈ (0, $∗ ), there exist C1 , γ0 , a0 , b0 > 0 such that, for each h ∈ Ωd0,1
(M ),
γ ∈ (0, γ0 ), z = a + ib with a ≥ a0 + σds , |b| ≥ b0 and n ≥ C1 ln |b|, we have
∗
b
Rn (z)h ≤
η
C#
|b|−γ khk∗η .
n
(a − σds )
59
7.1 Dolgopyat’s estimate
7 CONTACT FLOWS
Proof. First note that, for each v ∈ V u , Lv φ∗−t h = φ∗−t L(φ−t )∗ v h and, by the
Anosov property, k(φ−t )∗ vkC 0 ≤ C# e−λt kvkC 0 .
kLt hku ≤ C# e−λt sup kLt (Lv h)k0,0,ds ≤ C# e(htop (φ1 )−λ)t khku .
(7.2)
v∈V u
Thus
u
b
u
Rn (z)h ≤ C# (a − σds + λ)−n khk .
(7.3)
The other part of the norm is much harder to study. Let us start by noticing
that the proof of the second inequality in Lemma 4.3 can be used, as in
Lemma 4.6, to yield
s
C#
C#
b
khksη +
khks1+η .
(7.4)
Rn (z)h ≤
n
n
(a − σds + λ)
(a − σds )
η
s
It is then sufficient to consider the case h ∈ Ωd0,r
, g ∈ Γ`,1+η
. In the following
0
(`)
we will set hs = Ls h. It is convenient to proceed by small time steps of size
r > 0, to be fixed later. Let p(t) = p̃(δt) where p̃ is as in the proof of Lemma
6.7. For n ∈ N large enough, we must estimate (see (4.10))
Z ∞ Z
X Z ∞ tn−1 e−zt p(k − tr−1 )
tn−1 e−zt
∗
dt
dt
hg, φ−t hs i =
(−1)`(d−`) (n − 1)!
ca n
Wα,G (n − 1)!
k∈N ca n
(7.5)
Z
∗
×
h∗φt ∗ g, hs i ◦ φ−t+kr JW φkr Jφ−t ◦ φkr .
φ−kr Wα,G
Next, it is convenient to localize in space as well. To this end we need to
define a sequence of smooth partitions of unity.
1
For each i ∈ ZdQ
, let us define xi = rd− 2 i. We then introduce the partition
1
of unity Φr,i (x) = dl=1 p(d 2 r−1 xil − il ) (limited to the set Bd (0, 30δ)). Note
that it enjoys the following properties
(i) Φr,i (z) = 0 for all z 6∈ Bd (xi , r);
(ii) the collection {Bd (xi , r)}i∈Bd (0,10δ) covers Bd (0, 30δ) with a uniformly
bounded number of overlaps;
(iii) for each r, i we have k∇Φr,i kL∞ ≤ C# r−1 .
60
7.1 Dolgopyat’s estimate
7 CONTACT FLOWS
Note that the collection in (ii) has a number of elements bounded by C# r−d .
For each k ∈ N let Wα,G,β,kr be the family of manifolds defined in (4.9),
with t = kr, and set Wα,G,β,kr,i = {W ∈ Wα,G,β,kr : Θβ (W ) ∩ Bd (xi , r) 6= ∅}.
.
For simplicity, let us adopt the notation Wk,β,i = Wα,G,β,kr,i . For W ∈ Wk,β,i
.
f=
let τW : W
∪t∈[−2r,2r] φt W → R be defined by φ−τW (x) (x) ∈ W . Also set
ϕk,β,i (x) = ψβ (x)Φr,i (Θβ (x))p(r−1 τW (x))kV (x)k−1 .
(7.6)
Letting
(kr − τW )n−1 JW φkr ◦ φ−τW JφτW
∗ φ∗kr−τW ∗ g,
(−1)`(d−`) e−z(kr−τW ) Jφkr (n − 1)!
ĝk,β,i = ϕk,β,i
we can rewrite (7.5) as59
Z ∞ Z
tn−1 hg, φ∗−t hs i X
dt
=
ezt (n − 1)!
Wα,G
cn
k,β,i
X
W ∈Wk,β,i
(7.7)
Z
hĝk,β,i , hs i.
(7.8)
f
W
Recalling (4.16), we have
JW φkr ◦ φ−τ Jφ−kr+τ ∗ φ∗kr−τ ∗ g `,1+η
≤ C# .
W
W
W
Γ
(W )
(7.9)
Moreover, the derivative of ĝk,β,i in the flow direction is uniformly bounded
.
i
as well. Next, for each β, i we choose the reference manifold Wδs (Θ−1
β (x )) =
i
f
W β,i ∈ Σs , and define Wβ,i = W β,i ∩Θ−1
β (Bd (x , 2r)) and Wβ,i = ∪t∈[−2r,2r] φt (Wβ,i ).
.
Let us set HW
fβ,i ,W
f = Hβ,i,W . We can then rewrite the right hand side of (7.8)
multiplied by (−1)`(d−`) as60
X X Z
JHβ,i,W · hĝk,β,i , hs i ◦ Hβ,i,W
k,β,i W ∈Wk,β,i
=
X
fβ,i
W
X
k,β,i W ∈Wk,β,i
Z
hĝk,β,i,W , hs i + O
fβ,i
W
59
r2 (kr)n−1 e(σds −λ)s khku
(n − 1)!e(a−σds )kr
(7.10)
,
f
First, for each k ∈ N, perform the change change of variables Lk : R × W → W
defined by Lk (t, y) = φ−t+kr (y) and then introduce the partition of unity in space. Note
f appears several
that this implies t = kr − τW (x). Also, note that the same manifold W
times. Yet, given the support of the functions, the value of the integral is the correct one.
60
Form now on we can use the norms on the manifolds and the norms in the charts
interchangeably, since they are proportional.
61
7.1 Dolgopyat’s estimate
7 CONTACT FLOWS
where, in the last line, we have used Lemma 7.4, equation (7.2) and set
X
ĝk,β,i,W =
JHβ,i,W · ĝk,β,i,j̄ ◦ Hβ,i,W · ωβ,j̄ .
(7.11)
j̄
It is then natural to define
.
gk,β,i =
X
ĝk,β,i,W .
(7.12)
W ∈Wk,β,i
Recalling equations (7.6) (7.9), and Appendix D one can easily compute61
kgk,β,i k
Cη
≤ C# r
−η (kr)
n−1 −akr
e
(n − 1)!
· #Wk,β,i .
(7.13)
To conclude we need an estimate of the sup norm of the test forms and this
will follow from an L2 estimate,62
Z
2+ ds +1
− ds +1
kgk,β,i k2 ≥ C# kgk,β,i kC 0 η kgk,β,i kC η η .
fs
W
β,i
Hence
2η
ds +1
2η+ds +1
s +1
kgk,β,i kC 0 ≤ C# kgk,β,i kL2η+d
.
2 (W
f s ) kgk,β,i kC η
(7.14)
β,i
To compute the L2 norm it is convenient to proceed as follows. Given W ∈
. f
i
Wk,β,i let {xW } = W
∩Θ−1
β ({x +(0, η)}η∈Bdu (0,r) ) be its “central” point. Also,
e ρu (W ) = {Θβ (xW ) + (0, u, 0)}u∈B (0,ρ) .
for each ρ > 0, consider the disks D
du
Then, for a fixed % ∈ (0, r) to be chosen later, let Ak,β,i (W ) = {W 0 ∈ Wk,β,i :
f0 ) ∩ D
e u (W ) 6= ∅} and Bk,β,i (W ) = Wk,β,i \ Ak,β,i (W ) and set
Θβ (W
%
.
Gk,β,i,A =
X
W ∈Wk,β,i
Gk,β,i,B
.
=
X
W ∈Wk,β,i
X
W 0 ∈A
hĝk,β,i,W , ĝk,β,i,W 0 i
k,β,i (W )
X
W 0 ∈B
hĝk,β,i,W , ĝk,β,i,W 0 i,
k,β,i (W )
61
By #A we mean the cardinality of the set A.
Note that if f : Rν → R+ is a η-Hölder function and x0 satisfies f (x0 ) = kf kC 0 ,
then f (x) ≥ kf kC 0 − kf kC η |x0 − x|η , thus f (x) ≥ 21 kf k∞ for all x ∈ Bν (x0 , ρ) where
1
ρ = [kf kC 0 (2kf kC η )−1 ] η .
62
62
7.1 Dolgopyat’s estimate
7 CONTACT FLOWS
note that hgk,β,i , gk,β,i i = Gk,β,i,A +Gk,β,i,B . We will estimate the two terms by
separate arguments, the first being similar to [30, Lemma 6.2],63 the second
being the equivalent of [30, Lemma 6.3].
n−1 e−ark
2η
#Wk,β,i , p = 2η+d
.
To simplify the notation we set: Dk,β,i = (kr)(n−1)!
s +1
2
Lemma 7.7. If % < δr(1+ς) $̂ , for some ς > 0, then there exists ς0 > 0 s.t.
2
kGk,β,i,A k∞ ≤C# Dk,β,i
r ς0 .
(7.15)
Proof. Note that the number of elements in Ak,β,i (W ) must correspond to
the number of intersections between ∪t∈[−3δ,3δ] Wα,G and φkr (D%u (W )) and,
since each intersection has a δ-neighborhood in φkr (D%u (W )) that cannot
contain any other intersection, #Ak,β,i (W ) ≤ C# vol(φkr (D%u (W ))). On the
other hand, by the mixing property,64 a disk in φkr (D%u (W )) with inner diameter larger than C# must intersect ∪t∈[−3δ,3δ] Wα,G , thus #Ak,β,i (W ) ≥
C# vol(φkr (D%u (W ))). Similar arguments imply that #Wk,β,i is proportional
to vol(φkr (Dru (W ))). We are thus reduced to estimating the ratio of two
volumes.
The simplest possible estimate is as follows:65 let n be the largest integer
such that eλn r ≤ r−ς , now suppose that eλ+ n % ≤ δ, it follows that at time
n the image of the smaller disk is contained in a disk of radius δ while the
image of the largest one contains at least r−ςdu such disks. By the results
in Appendix C such a ratio will persist at later times. Note that the above
2
condition is implied by % < δr(1+ς) $̂ .
0
0
Lemma 7.8. Setting |b| = %−2+$ +ς and r = %1−$ +ς , for ς > 0, there exists
γ0 > 0 such that
Z
2
Gk,β,i,B ≤ C# b−γ0 rds +1 Dk,β,i
.
(7.16)
W
fs
β,i
The proof of the above Lemma is postponed to Subsection 7.2.
63
But note that in the present case we obtain much weaker results. The difficulty stems
from the fact that the ambient measure is the measure of maximal entropy rather then
the Liouville measure.
64
Recall that a contact flow on a connected manifold is mixing, [26].
65
It may be possible to improve on this by using some distortion estimate but we do
not see how.
63
7.1 Dolgopyat’s estimate
7 CONTACT FLOWS
Remark 7.9. Note that the conditions of the two Lemmata can be simultaneously satisfied only if $̂ > 23 .
Substituting in (7.14) the estimates given by (7.13), (7.15) and (7.16) we
obtain that there exists γ > 0 such that66
kgk,β,i kC 0 ≤ C# Dk,β,i |b|−γ .
(7.17)
By (7.13) and (7.17) and arguing as in the proof of (4.18), provided s ≥ ca n
with n ≥ C1 ln |b|, for C1 large enough,67 we can rewrite the last integral in
(7.10) as
X Z
X
fβ,i khks
Dk,β,i (b−γ + e−ληs r−η ) vol φ−s W
hgk,β,i , hs i ≤ C#
η
fβ,i
W
k,β,i
k,β,i
P
fβ,i | ≤ C# r P
Note that i #Wk,β,i |φ−s W
W ∈Wα,G,β,kr |φ−s W |. Thus, by Lemma
C.3 and Remark C.4,
X Z
X (kr)n−1 (|b|−γ + e−ληs r−η )
hgk,β,i , hs i ≤ C#
r khksη
ark−σ
(kr+s)
d
s
(n − 1)!e
fβ,i
W
k
k,β,i
(7.18)
≤ C# (a − htop (φ1 ))−n |b|−γ ehtop (φ1 )s khksη .
Thus, by (7.8), (7.10), (7.18),
s
b
2 R
(z)
h
n
1+η
khku (a − σds )n
C#
s
−γ
≤
+ |b| khkη .
(a − σds )n (a − σds + λ)n
(7.19)
Next, by (7.4), (7.19)
s
b
3 R
(z)
h
n
≤
η
≤
C#
(a − σds )3n
s
b
2 C# Rn (z) h
η
n
λ)
(a − σds +
(a − σds )n
(a − σds + λ)n
s
C#
b
2 R
(z)
h
n
(a − σds )n
1+η
+ |b|−γ khk∗ .
+
The above, and (7.3), imply the Lemma by choosing n appropriately.
66
67
Since, for all α, β > 0, p ∈ (0, 1), α1−p β p ≤ α + β.
So that the size of φ−s Wβ,i exceeds δ.
64
(7.20)
7.2 A key (but technical) inequality
7 CONTACT FLOWS
The goal of this section easily follows.
Lemma 7.10. If φt is a contact Anosov flow with $0 > 23 , then for each
η ∈ (0, $∗ ), there exist constants C1 , γ0 , a0 , b0 > 0 such that for each η ∈
(0, $∗ ), h ∈ B 1,η,ds , γ ∈ (0, γ0 ), z = a + ib with a ≥ a0 + σds , |b| ≥ b0 , and
n ≥ C1 ln |b|, holds true
|||R(z)n h|||1,η,ds ≤ Cη (a − σds )−n b−γ |||h|||1,η,ds .
Proof. By Lemma 4.6 we have
R(z)(k+1)n h
1,η,ds
≤
C#
kn
) (a −
|||h|||1,η,ds
(a − σds
σds + λ)n
C# |b| kn +
R(z)
h
.
0,1+η,ds
(a − σds )n
Recalling (7.1) and subsequent comments, it suffices to prove, for some k ∈ N,
b
≤ Cq (a − σds )−kn b−1−γ khk1,η,ds .
Rn (z)k h
0,1+η,ds
By Lemmata 3.7, 7.5 and Proposition 7.6 we have
C# ε khk1,η,ds
b
k
≤
+ C# Rn (z) Mε h
kn
(a − σds )
0,1+η,ds
0,1+η,ds
∗
C# ε khk1,η,ds
b
k
≤
+ C# R
n (z) Mε h
kn
(a − σds )
η
C# ε khk1,η,ds
≤
+ C# (a − σds )−kn |b|−kγ kMε hk∗η
kn
(a − σds )
C# ε khk1,η,ds
≤
+ C# (a − σds )−kn |b|−kγ ε−1 khk0,η,ds ,
(a − σds )kn
b
k Rn (z) h
which, after choosing ε = |b|−1−γ , n = C1 ln |b| and k = 2γ −1 + 2, proves the
Lemma.
7.2
A key (but technical) inequality
We are left with the task of proving (7.16).
65
7.2 A key (but technical) inequality
7 CONTACT FLOWS
Proof of Lemma 7.8. This is the heart of Dolgopyat’s estimate. Given
W ∈ Wk,β,i and W 0 ∈ Bk,β,i (W ), we must estimate
Z
Z
hĝk,β,i,W , ĝk,β,i,W 0 i =
JHβ,i,W · JHβ,i,W 0 · ĝk,β,i,ī ◦ Hβ,i,W
fs
fs
W
W
(7.21)
β,i
β,i
× ĝk,β,i,ī ◦ Hβ,i,W 0 .
It turns out to be convenient to write the above as an integral over W . More
precisely, let ϑ̃W : Bd (0, δ) → Rd be a flow box coordinate change preserving
the contact form α0 and such that, setting ϑW = Θ−1
β ◦ ϑ̃W , has the property
that ϑW ({(ξ, 0)}ξ∈Bds +1 (0,C# r) ) ⊂ W and, more, contains the support of the
integrand projected on W via the unstable Holonomy.68 We can then rewrite
(7.21) as69
Z
Z
e
hĝk,β,i,W , ĝk,β,i,W 0 i = dξ e−z(τW 0 ◦HW 0 (ξ)−ξd ) Gk,β,i,W,W 0 ,ī (ξ),
(7.22)
fs
W
β,i
Bds +1 (0,2r)
˜ ξd ), H
e W 0 = Hf f0 ◦ ϑW , and
where ξ = (ξ,
W ,W
Gk,β,i,W,W 0 ,ī
)
(
(kr + τW )n−1 e−akr JW φkr+τW
.
(∗φ∗kr+τ ∗ g)ī ◦ ϑW
= ϕk,β
(n − 1)!
Jφkr+τW
(kr + τW 0 )n−1 e−akr JW 0 φkr+τW 0
∗
eW 0
× ϕk,β
(∗φkr+τW 0 ∗ g)ī ◦ H
(n − 1)!
Jφkr+τW 0
e β,i,W 0 .
× JH
Let us define
.
e β,i,W 0 (ξ) − ξd ,
∆∗ (ξ) = τW 0 ◦ H
(7.23)
and note that ∆∗ (ξ + ζ) − ∆∗ (ξ) is exactly the so called temporal function.70
e
We let w(ξ) = (ws (ξ), wu (ξ), wd (ξ)) = ϑ−1
W ◦ HW 0 (ξ) − (ξ, 0). Then by Lemma
68
Such a coordinate change always exists, see [5, Lemma A.4].
f by the unstable holonomy is
By Appendix D it follows that the image of W 0 on W
strictly contained in a ball of radius 2r, provided r is small enough.
70
In the language of [30] (in which the role of the stable and unstable manifolds are
e W 0 (ξ) and y 0 = ϑW (ξ +
reversed) ∆∗ (ξ + ζ) − ∆∗ (ξ) = ∆(y, y 0 ), where x = ϑW (ξ), y = H
(ζ̃, 0)) with ζ = (ζ̃, ζd ).
69
66
7.2 A key (but technical) inequality
7 CONTACT FLOWS
D.2 we have71
0
0
|∆∗ (ξ + ζ) − ∆∗ (ξ) − dα0 (w(ξ), ζ)| ≤C# kwk2 kζ̃k$ + C# kwk$ kζ̃k2
0
0
+ C# kwk1+$ kζ̃k1+$ .
(7.24)
We are left with the task of performing the integral in the lefthand side of
(7.22). The basic idea is to integrate first in a specially chosen (optimal)
direction. As such a direction can change wildly from point to point (at least
in the case of very low regularity of the foliations) we impose the condition
0
r = %1−$ +ς
(7.25)
for some ς > 0. By Lemma D.1 we have that
0
0
kw(ξ) − w(ξ + ζ)k ≤ C# (kw(ξ)kkζ̃k$ + kw(ξ)k$ kζ̃k) ≤ C# %.
(7.26)
Next, we consider the directions yi = (−w(0), 0)kw(0)k−1 . By the cone
condition, kws k ≤ C# kwu k. Hence, for ξ ∈ Bds +1 (0, r), kw(ξ)k ≥ %,
dα0 (w(ξ), yi ) ≥ C# kwu (ξ)k.
(7.27)
For each ξ ∈ Bds +1 (0, r), hξ, yi i = 0, we define Ii (ξ) = {s ∈ R : ξ + syi ∈
Bds +1 (0, r)} and consider the integral
Z
∗
−z∆ (ξ+syi )
ds
e
G
0 ,ī (ξ + syi ) .
(7.28)
k,β,i,W,W
Ii (ξ)
It is then natural to divide the interval Ii (ξ) in subintervals {sl , sl+1 }l∈Z such
that,
(sl+1 − sl )dα0 (w(ξ + sl yi ), yi ) = 2π|b|−1 .
Let us set δl = sl+1 − sl and wl = w(ξ + sl yi ), then, recalling (7.27), we have
C# |b|−1 ≤
δl kwl k
≤ C# |b|−1 .
2π
(7.29)
Next, we must assume
71
It may be possible to improve this estimate by going forward or backward in time to
the situation in which the two lengths are equal, but some non-obvious distortion estimate
would have to play a role to insure that the following “preferred” direction does not changes
too wildly.
67
7.2 A key (but technical) inequality
7 CONTACT FLOWS
|b|−1 ≤ r%,
(7.30)
to insure that Ii (ξ) contains at least an interval. Note that Lemma D.1 ime W 0 (ξ + yi sl+1 ) − H
e W 0 (ξ + yi sl )k ≤ C# max{δl , kwl kδ $0 }. Hence72
plies that kH
l
2
sup |Gk,β,i,W,W 0 ,ī (ξ + syi ) − Gk,β,i,W,W 0 ,ī (ξ + sl yi )| ≤ C# Dn,k
|s−sl |≤δi
0
× (r−1 max{δl , kwl kδl$ } + δl$∗ kwl k)
. (kr)n−1 e−akr
where Dn,k =
.
(n − 1)!
We can then bound (7.28) by
X Z sl+1 ∗
∗ (ξ+s y )]
−z[∆
(ξ+sy
)−∆
i
l i
e
G
0 ,ī (ξ + sl yi )
k,β,i,W,W
sl
l
+
2
C# Dn,k
kwl k1−$∗ r
−1
−1
−$0
1−$0
|b| kwl k + |b| kwl k
.
+
|b|$∗
Next, equations (7.24), (7.29) yield
0
kwl k2−$
|∆ (ξ + syi ) − ∆ (ξ + sl yi ) − (s − sl )dα0 (wl , yi )| ≤ C#
|b|$0
0
1
kwl k$ −2
+
C
.
+ C#
#
|b|2
|b|1+$0
∗
∗
Recalling (7.25) and (7.26), we can continue our estimate of (7.28) as
X Z sl+1
−z(s−sl )dα0 (wl ,ȳl )
ds e
Gk,β,i,W,W 0 ,ī (ξ + sl yi )
sl
l
0
0
0
r1−$∗
r2−$
%$ −2 r−$
2
−1 −1 −1
+ C# Dn,k r |b| % r + $∗ + $0 −1 +
+ $0 .
|b|
|b|
|b|
|b|
We remark that the integrals in the first line are all bounded by C# |b|−1 δl .
By the above equation, integrating in the remaining directions and letting73
.
Dk,n,β,i = Dk,n #Wk,β,i and
72
73
0
|b|−1 = %2−$ +2ς
e β,i,W 0 (ξ) − J H
e β,i,W 0 (ξ + ζ)k ≤ C# kwl kkζk$∗ , see Appendix D.
We remark that kJ H
The choice for % satisfies the constraints (7.30) .
68
8 GROWTH OF ζ-FUNCTIONS
equation (7.22) yields
Z
"
#
$0 ς
1
0
2
2
%$ ς + $0 rds +1 ≤ C# Dk,n,β,i
|b|− 2−$0 +2ς rds +1 .
Gk,β,i,B ≤ C# Dk,n,β,i
W
fs
|b| 2
β,i
which gives the required inequality.
8
Growth of ζ-functions
We start by showing that the estimates in section 7.1 imply a bound on the
growth of the traces. To this end note that the previous results show that
tr[ (R(z)n ), which is well defined for <(z) large enough, equals a function
, for each p + q < r − 1. It is then natural
meromorphic in <(z) > σ` − σp,q
2
[
n
to use tr (R(z) ) to also denote such an extension.
Lemma 8.1. There exists τ∗ , B > 0, A ≥ htop (φ1 ), such that, for each n ∈ N,
z ∈ C, 2A ≥ a = <(z) ≥ A, b = =(z) 6∈ [−B, B],
| tr[ (R(z)n )| ≤ C# |b|γ (a − σds + τ∗ )−n .
Proof. By Lemma 5.1 and equation (5.10) follow, for each L > 1 and ε ≤ εL ,
Z
X
[
n
(`)
n
n −C# Ln
tr (R(z) ) ≤ + C# ε(a − C2 )−n + C#
hω
,
R
(z)
(j
)i
e
x
α,
ī
,α,
ī,x
M α,ī
n −C# Ln
≤ C# ε−C# |||R(z)n |||1,q,ds + C# ε(a − C2 )−n + C#
e
Next, given b ∈ R, large enough, let z = a + ib, a = a0 + σds and c > 0.
Also, let nb = dC1 ln |b|e and write, for any n ∈ N, n = kn nb + rn , rn < nb .
Lemmata 7.10 and 4.6 imply, for B large enough,
n
n
|||R(z) |||1,q,ds
.
Let us set C̄a =
−C#
γ − γ2 n
γ
Cqkn |b|−γkn
|b|γ e ln |b| (ln Cq −γ ln |b|)
|b|γ e− 2 n
≤
≤
.
≤
(a − σds )n
(a − σds )n
(a − σds )n
C2 +σds
.
a−σds
(8.1)
We can then chose ε depending on n by requiring
= C# εe . This, provided A is large enough and e−C# L =
C# ε
|b| e
(2A − σds + 1)−1 yields
[
tr (R(z)n ) ≤ C# (a − σds )−n |b|γ e− γ4 n + C# (a − σds + 1)−n
C̄a n
implying the result.
69
A EXTERNAL FORMS: A TOOLBOX
Lemma 8.2. In the region {z ∈ C : <(z) ∈ [htop (φ1 ) − τ∗ , ∞)} ζRuelle
is non zero and has only a simple pole at z = htop (φ1 ). Moreover, setting
D = {z ∈ C : <(z) ∈ [htop (φ1 ) − τ∗ , A] ; |=(z)| ≥ B} we have
γ
|ζRuelle (z)| ≤ C# eC# |z| 1
d
ln ζRuelle (z) ≤ C# |z|γ1
dz
for all z ∈ D
for all z ∈ D.
Proof. If τ∗ is chosen small enough, then all the determinants D` , ` 6= ds , are
analytic and bounded in the considered regions and hence, recalling equation
(2.3), it suffices to study Dds (z). In view of Lemma 2.10 we start by studying
e ds (ξ−z, ξ). Fix z0 ∈ D and consider ξ = A+i=(z0 ). Then, recalling Lemma
D
5.1
!
∞
X
(ξ − z)n [
Dds (z) = Dds (ξ)D̃ds (ξ − z, ξ) = Dds (ξ) exp −
tr (R(ξ)n ) .
n
n=1
Note that Dds (ξ) is uniformly bounded in z0 . By Lemma (8.1) it follows that
!
∞
X
|=(z0 )|γ (A − <(z0 ))n
|Dds (z0 )| ≤ C# exp −
n(A − htop (φ1 ) + τ∗ )n
n=1
which is convergent provided <(z0 ) > htop (φ1 ) − τ∗ .
Next, by the same argument as above, the logarithmic derivative
∞
X
d
(ξ − z)n−1 tr[ (R(ξ)n ).
ln Dds (z) =
dz
n=1
is bounded as claimed in the required domain.
Appendices
A
External Forms: a toolbox
In this appendix we collect together for the reader’s convenience some useful
formulae. Most of them are a direct application of [24, Section 2]).
70
A EXTERNAL FORMS: A TOOLBOX
Given a Riemannian d dimensional manifold M , for each x ∈ M , ` ∈
{0, . . . , d} and v1 , . . . , v` , w1 , . . . , w` ∈ Tx∗ M we define74
hv1 ∧ · · · ∧ v` , w1 ∧ · · · ∧ w` i = det(hvi , wj i).
(A.1)
Assuming bilinearity, the above formula defines uniquely a scalar product
among `-forms.
Let ωM be the Riemannian volume form on M . We define a duality from
` to d − ` forms via (see [24, (2.1.6)])
hv, wiωM = (−1)`(d−`) v ∧ ∗w = (−1)`(d−`) w ∧ ∗v = ∗v ∧ w.
(A.2)
Since such a formula must hold for each `-forms, the (d − `)-form ∗w is
uniquely defined. The operator “∗” is the so called Hodge operator. In
particular, equation (A.2) implies the following equalities
hv, ∗wi = (−1)`(d−`) h∗v, wi ; ∗ ∗ v = (−1)`(d−`) v ; h∗v, ∗wi = hv, wi. (A.3)
It is also natural to define a scalar product
Z
.
hf, giΩ` =
hf, gix ωM (x).
M
Next, consider a smooth diffeomorphism F : M → N , for M, N Riemannian
manifolds. Let det(DF ) be determined by F ∗ ωN = det(DF )ωM then
hF ∗ f, gi = (−1)(d−`)` det(DF ) · F ∗ hf, ∗(F −1 )∗ ∗ gi.
(A.4)
To prove this, we just compute
hF ∗ f, giωM = ∗ ∗ F ∗ f ∧ ∗g = F ∗ (f ∧ ∗ ∗ (F −1 )∗ ∗ g)
= (−1)`(d−`) F ∗ (hf, ∗(F −1 )∗ ∗ giωN )
= (−1)`(d−`) det(DF ) · hf, ∗(F −1 )∗ ∗ gi ◦ F · ωM .
In particular, letting (F ∗ )0 be defined by hF ∗ v, wiΩ` = hv, (F ∗ )0 wiΩ` , we have
∗(F ∗ )0 = (F −1 )∗ ∗ .
74
By duality the scalar product in T∗ M induces a canonical scalar product in T ∗ M .
71
B ORIENTABILITY
The above formulae yield a formula for the Lie derivative: let Z be a vector
field and Ft the flow generated by Z. Then, by differentiating Ft∗ hv, wi =
det(DF )−1 hFt∗ ∗ v, ∗Ft∗ wi with respect to t at t = 0, it follows that
LZ hv, wi = −hv, wi div Z + hLZ (∗v), ∗wi + hv, LZ wi
= −hv, wi div Z + (−1)d(d−`) h∗LZ (∗v), wi + hv, LZ wi.
(A.5)
We will also be using the relation (see [1, Proposition 2.2.19] for the details)
LZ F ∗ w = F ∗ LF∗ Z w.
(A.6)
In addition, note that if {dxi } is an orthogonal base of T ∗ M , then {dxī =
dxi1 ∧ · · · ∧ dxi` } is an orthonormal base of ∧` T ∗ M , where ī = (i1 , . . . , il ) is
an ordered multi-index (i.e. ik < ik+1 ). Then
∗dxī = ε(īc )dxīc
where īc is the ordered (d − `)-multiindex such that dxī ∧ dxīc = ε(ī)ωM and
ε(ī) is the sign of the permutation π(1, . . . , d) = (i1 , . . . , i` , ic1 , . . . , icd−` ).
Let γV be a rescaling of the vector field V . Recall that, by definition,
.
X (`) h = dtd φ∗−t ht=0 = −LV h for h ∈ Ω`0,s . Hence for all t ∈ R, by the
properties of Lie derivatives (for iV as in equation (3.14)),
(`)
(`)
(`)
LγV (Lt h) = γLV (Lt h) + dγ ∧ iV (Lt h)
d ∗
(`)
∗
= −γ φ−s (φ−t h)
= −γLt (X (`) h).
ds
s=0
B
(A.7)
Orientability
Here we prove some simple facts about orientability of the invariant distributions for Anosov flows and discuss how to modify the arguments of this
paper in the non orientable case. We believe some of the following results to
be well known but we could not locate a simple reference, so we add them
for the reader convenience.
Lemma B.1. For each geodesic flow on an orientable compact Riemannian
manifold M0 of negative sectional curvature, the unstable and stable distribution are orientable.
72
B ORIENTABILITY
Proof. We present the proof for the unstable distribution, the proof for the
stable distribution being exactly the same by reversing time. Remember that
.
the geodesic flow takes place in M = T1 M , the unitary tangent bundle. If
we assume that M0 is d dimensional, then M is 2d − 1 dimensional and the
unstable space is d − 1 dimensional.
Given a geodesic γ, let (J, J 0 ) ∈ T (T M ) be a Jacobi fields along it. We
have then [13, Chapter 5.2]
D2 J
+ R(γ 0 (t), J(t)))γ 0 (t) = 0
dt2
D
, is the covariant derivative and R is the curvature tensor. We can
where dt
D
recall that dt
J = J 0 and that we can assume that both J(t) and J 0 (t) are
perpendicular to dtd γ(t). If we then define the quadratic form Q(J, J 0 ) =
hJ, J 0 i, we have [13, chapter 3.3],75
d
Q(J, J 0 ) = hJ 0 , J 0 i − hJ, R(γ 0 (t), J(t))γ 0 (t)i
dt
= hJ 0 , J 0 i − K(γ 0 (t), J(t)) kγ 0 (t) ∧ J(t)k2 > 0.
This means that the cone {(J, J 0 ) ∈ T (T M ) : Q(J, J 0 ) > 0} is invariant
under the flow, thus the unstable vectors must belong to such a cone. Given
a point q ∈ M0 and (q, p) ∈ M , let us consider the tangent space T(q,p) M .
Let (δq, δp) ∈ E u (q, p) ⊂ T(p,q) M . By the above discussion we have hδq, pi =
hδp, pi = 0 and that (0, δp) ∈ E u implies δp = 0. Finally, let E cu (q, p) =
span{E u (q, p), V }, where V = (p, 0) is the vector field generating the geodesic
flow. Accordingly, if we denote the canonical projection π : M → M0 defined
by π(q, p) = q, we have that π∗ : E cu (q, p) → Tq M is an isomorphism. Indeed,
if (δq, δp) ∈ E cu , then we can write δq = βp + ξ and δp = η, where β ∈ R,
and (ξ, η) ∈ E u . Accordingly, if δq = π∗ (δq, δp) = 0, then β = 0 and ξ = 0,
which in turns imply η = 0.
To conclude note that since M0 is orientable, then there exists a volume
form ω on M0 . But then ω = iV (π ∗ ω) is a volume form on each E u , therefore
the bundle is orientable.
The above result is false for general Anosov flows.76 Yet, it is often
possible to keep track of the orientation of an orbit by a simple multiplicative
factor as follows.
75
76
We let K denote the sectional curvature.
For example consider the map f : T2 → T2 defined by f (x, y) = −A(x, y) mod 1
73
B ORIENTABILITY
Lemma B.2. If the torsion of the first homology group does not contain
factors of the type Z2m , then there exists a smooth function A : M × R+ → R
such that, for each closed orbit τ and x ∈ τ ,
eπiA(x,λ(τ )) = (τ ).
Moreover, At (·) = A(t, ·) is a cocycle.
Proof. Consider the line bundle π : F → M such that the fiber at x ∈ M
consists of the volume forms on E u (x). Let F0 = F \ {(x, 0)}x∈M , i.e. we
have taken out the zero section. Hence, for each x ∈ M , π −1 (x) has two
connected component. Thus, for each x ∈ M , there exists a neighborhood
U 3 π −1 x and a continuous map F : U → π(U ) × Z2 such that F is constant
on each connected component and p ◦ F (x̄) = π(x̄), where p(x, i) = x. This
construction defines a double covering Mor of M . Given a cycle γ : [0, 1] →
M we can then consider any lift γ̃ to Mor . Locally γ̃ will have the form
¯ = i(0) + i(1)
γ̃(t) = (γ(t), i(t)), i ∈ Z2 . We define then the degree map d(γ)
which is well known to be a homotopy invariant. Accordingly, d¯ depends
only on the homology class of γ. In other words d¯ ∈ H 1 (M, Z2 ). We would
like to show that d¯ ∈ H 1 (M, R).
Since H1 (M, Z) is finitely generated it is isomorphic to Zn ⊕ Tor. By the
universal coefficient Theorem H1 (M, R) is isomorphic to Rn while H1 (M, Z2 )
is isomorphic to Zn+m
where m is the number of Z2k that are present in the
2
torsion part. Thus by hypothesis m = 0. Next, it is known that the closed
orbits of an Anosov map generate all H1 (M, Z), [2]. Consider a set of closed
periodic orbits Γ = {γi }i∈N that generate H1 (M, Z), hence they generate
H1 (M, Z2 ). We can thus consider a base Γ0 ⊂ Γ of H1 (M, Z2 ). By the above
discussion it is a base in H1 (M, R) as well. We can then consider the dual
base in H 1 (M, R). By De Rham cohomology such a base can be represented
in terms of 1-forms. In conclusion, there exist closed 1-forms {wγ }γ∈Γ0 such
R λ(γ 0 ) ∗
that, for each γ, γ 0 ∈ Γ0 , 0
φt wγ (V ) is one if γ = γ 0 and zero otherwise.
Finally, let
Z t
X
¯
A(x, t) =
d(γ)
φ∗s wγ (V (x))ds.
0
γ∈Γ0
where
A=
1
1
1
.
2
One can then define a ceiling function τ and the associated suspension, clearly the resulting
flows is Anosov but the invariant distribution are non orientable.
74
B ORIENTABILITY
Note that A is an additive cocycle with respect to the flow φt . Moreover, if
¯ ).
τ is a closed orbit and x ∈ τ , A(x, λ(τ )) mod 2 = d(τ
In the hypotheses of the above Lemma we can define, instead of (2.7),
the operator
(`)
(B.1)
Lt (h) = eπiAt φ∗−t h.
The study of such an operator can be carried out in complete analogy with
what we have done in sections 4, 5, 6 (at the price of slightly heavier notation).77
If the hypothesis of Lemma B.2 does not apply, then one can still introduce an appropriate weight, but this creates some difficulties that can be
solved only by introducing a more sophisticated Banach space. The basic
idea is to consider the action of φ∗t also on an orientation bundle. This would
be simple enough, but in so doing the action of the flows on such a bundle
produces a multiplicative factor proportional to the stable Jacobian. To eliminate such an unwanted multiplicative factor it is then necessary to consider
a transfer operator with a weight given exactly by the inverse of the stable
Jacobian.78 Yet, this solution produces another problem: in general such
a weight is only Hölder, hence the resulting operator does not act properly
on spaces of smooth functions, a property essential to obtain large domains
of analyticity for the zeta function. To overcome this last obstacle in the
present setting one has to change the Banach space. This can be done as in
[20].
Let G be the Grassmannian of ds dimensional subspaces of the tangent
bundle.
consider the fiber bundle E := {(x, E, h, ω) : (x, E) ∈ G, h ∈
V` ∗ Next, V
Tx M, ω ∈ ds E 0 }, π : E → G given by π(x, E, h, ω) = (x, E). The fibers
consist of the space of `-forms times a line that gives a volume form (hence
an orientation) on the subspace. We can then consider the vector space S
of the C r−1 sections of the bundle E. This generalizes the space Ω`r . Given
(h, ω) ∈ S, g ∈ Ω`r and Wα,G ∈ Σ we can then define the integral
Z
Z
hg, hiω =
hg(Gα (ξ)), h(Gα (ξ), EGα )i G∗α ω
Wα,G
Rds
77
The computations are essentially the same. In Section 4, 6, 7 one obtains the same
bounds for the spectrum, while in Section 5 keeping track of the cocycle yields the wanted
factor (τ ).
78
The interested reader can consult Fried [17, Section 6] for a thorough explanation.
75
C TOPOLOGICAL ENTROPY AND VOLUME GROWTH
.
ds
where Gα = Θ−1
α ◦ G and EGα = DGα (R ). We can then use the above
to define the analogue of the functionals Jα,G,g,v1 ,...,vp in section (3.10) and
then construct the norms in the same way. The transfer operator is then the
canonical action of the flow on S with a weight given by the Jacobian of the
flow restricted to the element of the Grassmanian. Note that one obtains an
operator of the type considered in [20, equation (3.1)]. Accordingly, all the
present arguments can be repeated in complete analogy, although one must
compute in the more sophisticated way detailed in [20].79
As the present paper is already quite technical, we decided not to go into
such a more sophisticated construction as, on the one hand, it does not add
any new idea, and on the other it would make the presentation much harder
to follow. Nevertheless, all the results presented here can be generalized by
using such an approach.
C
Topological Entropy and volume growth
It is well known that there is a relation between asymptotic volume growth of
manifolds and the topological entropy (see [48, 18, 36] and references therein).
Unfortunately, here we need a uniform upper and lower bound for all times.
We are not aware of such bounds for flows in the literature. For Axiom A
diffeomorphisms they can be obtained in great generality from the results in
[20].80 It is clear that applying the same strategy to the present setting of
flows similar results can be obtained. Yet, since we are interested only in
the topological entropy and not in an arbitrary potential, a more elementary
approach is available.
Let us proceed in a slightly higher generality than needed, since it can
be done at no extra cost. Let M be a Riemannian manifold and (M, f ) a
partially hyperbolic diffeomorphism. Call E s , E u , E c the stable, unstable and
central distribution respectively. Let ds be the dimension of E s .
Assume that, for each n ∈ N, kDf n |E c k + kDf −n |E c k ≤ C# .
79
To be precise, in [20] there are no `-forms, but the point of the present paper is
exactly that functions and `-forms can be handled by essentially the same computations
once the proper machinery has been set up.
80
Indeed, in [20] it is shown in particular that, given a mixing Axiom A diffeomorphism (M, f ), the spectral radius
operator
with potential φ̄ satisfies, for each
R of aPtransfer
n−1
P (φ̄)n
φ̄◦f k P (φ̄)n
k=0
manifold W ∈ Σ, C# e
≤ We
, where P (φ̄) is the pressure
≤ C# e
associated to the potential, see [20, equations (3.3), (4.10), Lemma 4.7, Theorem 5.1.].
76
C TOPOLOGICAL ENTROPY AND VOLUME GROWTH
A set S ⊂ M is called (ε, n)-separated if for each x, y ∈ S we have that
df,n (x, y) > ε where df,n (x, y) = max0≤k≤n d(f k (x), f k (y)). Let S(f, ε, n) be
the set of all (ε, n)-separated sets. We then define N (f, ε, n) = sup #S.
S∈S(f,ε,n)
It is well known that, [27],
ln N (f, ε, n)
.
ε→0 n→∞
n
htop (f ) = lim lim
Moreover since if ε0 > ε, then N (f, ε, n) ≥ N (f, ε0 , n). Thus, for each δ > 0,
ln N (f, δ, n)
.
n→∞
n
htop (f ) ≥ lim
(C.1)
b be the set of all C 1 , ds -dimensional manifolds of
Fix δ > 0 and let Σ
radius δ and tangent space close to E s . Let us define
n
ρ+
n = sup vol(f W )
n
ρ−
n = inf vol(f W ).
;
W ∈Σ̂
W ∈Σ̂
Lemma C.1. For all n, m ∈ N,
− −
ρ−
n+m ≥ C# ρn ρm ;
+ +
ρ+
n+m ≤ C# ρn ρm .
In addition, if f is topologically transitive,
−
ρ+
n ≤ C# ρ n .
.
Proof. To prove the first equation consider f n (W ) and chose a set Wn =
b such that Wi ⊂ f n (W ), Wi ∩ Wj = ∅ and the cardinality of Wn is
{Wi } ∈ Σ
maximal. Then #Wn ≥ C# vol(f n (W )). Thus
X
−
vol(f n+m (W )) ≥
vol(f m (W 0 )) ≥ C# ρ−
n ρm
W 0 ∈Wn
and the result follows taking the inf on W . The second inequality is proven
similarly by taking a minimal cover of f n (W ).
Let us prove the last inequality. By topological transitivity for each ε > 0
there exists nε ∈ N such that given any two ball B, B 0 of radius ε, there exists
n ≤ nε such that f n (B) ∩ B 0 6= ∅.81 Let n̄ = nδ .
81
Take S ∈ S(f, 0, ε/4) with maximal cardinality. Then ∪x∈S B(x, ε) ⊃ M , where
B(x, ε) is a ball (in the Riemannian metric) or radius ε and center x. Then for each
x, y ∈ S there exists nx,y ∈ N such that f nxy (B(x, ε/4)) ∩ B(y, ε/4) 6= ∅. The claim
follows since any ball B of radius ε must contain a point x ∈ S such that B(x, ε/4) ⊂ B.
77
C TOPOLOGICAL ENTROPY AND VOLUME GROWTH
b such that vol(f n−n̄ (Wn )) ≥
Next, for each n ∈ N there exists Wn ∈ Σ
1 +
ρ . Let x ∈ Wn and B an open ball centered at x and of radius cδ for
2 n−n̄
b let z ∈ W and
some fixed c ∈ (0, 1) to be chosen later. Given any W ∈ Σ
0
B a ball of radius cδ centered at z. By the above arguments there exists
m ≤ n̄, such that f m (B 0 ) ∩ B 6= ∅. Given that the stable and unstable
manifolds are uniformly transversal, it is possible to choose c such that, for
each point in y ∈ Wn , Wδcu (y)∩f m W 6= ∅.82 By the hypothesis on the central
bundle it follows that the point related by the holonomy will never have a
distance larger than δ in the future, then vol(f k (Wn )) ≤ C# vol(f k+m (W )).
Accordingly
+
vol(f n (W )) ≥ C# vol(f n−m (Wn )) ≥ C# ρ+
n−n̄ ≥ C# ρn ,
and the result follows again by taking the inf on the W.
Lemma C.2. If (M, f ) is topologically transitive, then, for each n ∈ N,
ε > 0,
+
Cε ρ +
n ≤ N (f, δ, n) ≤ Cε ρn .
b and consider a ε separated set Sn on f n (W ).83 Since
Proof. Let W ∈ Σ
the manifold contracts backward, points that are far away in f n (W ) but
close in M will separate backward. Thus f −n Sn ∈ S(f, ε, n). Since #Sn ≥
C# vol(f n (W )), it follows that N (f, ε, n) ≥ Cε ρ+
n.
To prove the second inequality we use the notation introduced in the
b Let S ∈ S(f, ε, n) be a set with maximal
proof of Lemma C.1. Fix W ∈ Σ.
cardinality. Cover M with balls of radius cε. By transitivity, for each ball
B there exists a time nB ≤ n̄ such that f nB W ∩ B 6= ∅. Let WB ⊂ f nB W ,
b be a manifold that intersects the ball B and project all the point
WB ∈ Σ
in S ∩ B to WB via the weak unstable holonomy. If we consider the images
of such points we note that two points can be (ε, n) separated only if their
corresponding points on WB are (C# ε, n) separated. Finally, note again that
the (C# ε, n) separated points on WB must have a distance larger that C# ε
in the manifold f n (WB ). Thus there can be at most Cε vol(f n (WB )) such
points. Accordingly,
X
Cε vol(f n+nB (W )) ≤ Cε ρ+
N (f, ε, n) ≤
n.
B
82
By Wδcu (y) we mean a disk of radius δ in the weak unstable manifold.
That is the distance among the points is larger than ε when measured on the manifold
f n (W ).
83
78
D HOLONOMIES
Using the above results the required bound follows.
Lemma C.3. If (M, f ) is topologically transitive, then for all n ∈ N we have
htop (φ1 )n
.
ehtop (φ1 )n ≤ ρ+
n ≤ C# e
Proof. By Lemmata C.2, C.1
ln N (f, ε, kn)
ln Cε ρ+
ln Cε ln ρ+
n
kn
≤
≤
+
.
kn
kn
kn
n
Taking the limit for k → ∞ first and ε → 0 second, the first inequality of
the Lemma follows.
+
+ +
k
+ k
By Lemma C.1, it follows that ρ+
n+m ≥ C# ρn ρm . Then ρkn ≥ C# (ρn )
and
ln N (f, δ, kn)
ln ρ+
kn
≤ n lim
≤ nhtop (φ1 )
k→∞
k→∞ km
kn
ln[C# ρ+
n ] ≤ m lim
where we have used Lemma C.2 and equation (C.1).
Remark C.4. Note that if f is the time one map of a hyperbolic flow, then
by the spectral decomposition Theorem [11, Section 1.1], the manifold can
be decomposed into finitely many isolated topologically transitive sets. Hence
applying Lemma C.3 to each isolating neighborhood we obtain C# ehtop (φ1 )n ≤
htop (φ1 )n
ρ+
for each Anosov flow.
n ≤ C# e
D
Holonomies
The strong unstable foliation can be locally trivialized by a change of coordinates of the form H(ξ, η) = (H(ξ, η), η), η ∈ Bdu (0, δ), ξ ∈ Bds +1 (0, δ),
such that for each ξ, {H(ξ, η)}η∈Bdu (0,δ) is a strong unstable manifold. In
addition, without loss of generality, we can require that H(0, η) = 0 and
H(ξ, 0) = ξ. The results on the regularity of the foliation can be summarized by considering the map H̃ : Bds +1 (0, δ) → C 0 (Bdu (0, δ), Rds +1 ) defined by [H̃(ξ)](·) = H(ξ, ·). Then H̃ has range in C r (Bdu (0, δ), Rds +1 ) and
kH̃kC $ (Bds +1 (0,δ),C r (Bdu (0,δ),Rds +1 )) ≤ C# . Moreover for each η, the function
79
D HOLONOMIES
H(·, η) is absolutely continuous and has Jacobian JH(ξ, η). Again, we can define J H̃ : Bds +1 (0, δ) → C 0 (Bdu (0, δ), R) by [J H̃(ξ)](·) = JH(ξ, ·). Then J H̃
has range in C r (Bdu (0, δ), R) and kJ H̃kC $∗ (Bds +1 (0,δ),C r (Bdu (0,δ),R)) ≤ C# .84
In this paper the holonomies are often used as follows: given W1 , W2 ∈ Σ
85
and x, y ∈ W1 estimate d(HW
Although it is clear how
f1 ,W
f2 (x), HW
f1 ,W
f2 (y)).
to proceed, we give the details for the benefit of the lazy reader.
Note that, without loss of generality we can assume that x = 0, y = (a, 0)
f
f
and HW
f1 ,W
f2 (0) = (0, z), W1 = {(ξ, 0)}ξ∈Rds +1 . Then W2 will have the form
{(ξ, z + G(ξ)), G(0) = 0, and kDξ Gk ≤ 2. Also let HW
f1 ,W
f2 (a, 0) = (ξ, η),
Lemma D.1. We have that
h
i
0
k(ξ, η) − (a, z)k ≤ C# kak$ kzk + k∂ξ GkC 0 kak .
(D.1)
Proof. The intersection point (ξ, η) satisfies (ξ, η) = (H(a, η), z+G(ξ)). Thus
ξ − a = H(a, η) − H(a, 0)
= H(a, z + G(ξ)) − H(a, z + G(a)) + H(a, z + G(a)) − H(a, 0)
= A(ξ − a) + H(a, z + G(a)) − H(a, 0).
where the last line is an application of the intermediate value theorem and
0
kA(ξ)k ≤ k∂η H(a, ·)kC 0 k∂ξ GkC 0 ≤ C# kak$ k∂ξ GkC 0 . Then
0
kξ−ak ≤ k(1−A)−1 (H(a, z+G(a))−H(a, 0))k ≤ C# kak$ (kzk+k∂ξ GkC 0 kak).
Analogously,
η − z = G(H(a, η)) − G(H(a, z)) + G(H(a, z)) = A1 (η − z) + G(H(a, z)),
0
where kA1 (η)k ≤ k∂ξ GkC 0 k∂η H(a, ·)kC 0 ≤ C# kak$ k∂ξ GkC 0 . Thus
0
kη − zk ≤ k(1 − A1 )−1 G(H(a, z))k ≤ C# k∂ξ GkC 0 (kak + kak$ kzk).
Next, we need a small improvement of [30, Lemma B.7]. We use the same
notation established before (7.24), a part form the fact that the points ξ, ζ
are fixed to some value ξ∗ , ζ∗ .
84
85
See [23] and reference therein for the proofs of these statements.
f = ∪t∈I φt W , for some appropriate interval I.
Remember that W
80
REFERENCES
REFERENCES
Lemma D.2. For ξ∗ , ζ∗ ∈ Bds +1 (0, δ) and ∆∗ as in (7.23) we have that
0
0
|∆∗ (ξ∗ + ζ∗ ) − ∆∗ (ξ∗ ) − dα0 (z(ξ∗ ), ζ∗ )| ≤C# (kzk2 kζ̃∗ k$ + kzk$ kζ̃∗ k2 )
0
0
+ C# kzk1+$ kζ̃∗ k1+$ .
Proof. We use the same coordinates defined before Lemma D.1 with W =
f is a flat manifold and W
f0
W1 , W 0 = W2 ; so that ξ∗ = 0, ζ∗ = (ζ̃, 0).86 Thus W
˜ where ξ˜ = (ξ1 , . . . ξds ), ξds +1 being the flow direction.
is the graph of (ξ, G(ξ)),
˜
˜ G(ξ),
˜ L(ξ))
˜ with L(0) = 0,
Also W = {(ξ, 0)} while W 0 is the graph of (ξ,
G(0) = z̄ = z u (ξ∗ ). Let H(ξ, η) = (Hs (ξ, η), H0 (ξ, η)), Hs : Rd → Rds , H0 :
Rd → R, and {(Iu (ζ), Is (ζ), I0 (ζ))} = {H(ζ, η)η∈Rdu } ∩ {(ξ, G(ξ))ξ∈Rds +1 }.
Then ∆∗ (ξ∗ + ζ∗ ) − ∆∗ (ξ∗ ) = I0 (ζ̃) − L(Is (ζ̃)). Define Ξ : [0, 1]2 → Rd by
Ξ(t, s) = (sHs (ζ̃, tIu (ζ̃)), tG(sIs (ζ̃)), s[H0 (ζ̃, tIu (ζ̃)) − tI0 (ζ̃)] + tL(sIs (ζ̃)))
.
and set Ξ([0, 1]2 ) = Σ. From [30, equations (B.4), (B.5)] it follows that
I0 (ζ̃) − L(Is (ζ̃)) equals the integral of the symplectic form over Σ, hence
Z
Z
Z
∗
∗
∗
∆ (ξ∗ + ζ∗ ) − ∆ (ξ∗ ) =
dα0 =
Ξ dα0 =
dα0 (∂s Ξ, ∂t Ξ) dtds
Σ
[0,1]2
[0,1]2
Z
=
hHs (ζ̃, tIu (ζ̃)), G(sIs (ζ̃))i − tsh∂ξ̃ G(sIs (ζ̃)Is (ζ̃), ∂η Hs (ζ̃, tIu (ζ̃))Iu (ζ̃)i
[0,1]2
0
0
0
0
= hζ̃, z̄i + O kζ̃k$ kz̄kkIu k + kζ̃kkz̄k$ kIs k + kζ̃k$ kz̄k$ kIu kkIs k .
0
0
Since, by Lemma D.1, kIu − z̄k + kIs − ζ̃k ≤ C# (kz̄k$ kζ̃k + kz̄kkζ̃k$ ), the
Lemma follows.
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