In a series of two posts, we will revisit our previous discussion on the exponential mixing property for hyperbolic flows via a technique called Dolgopyat’s estimate.
Here, our main goal is to provide a little bit more of details on how this technique works by offering a “guided tour” through Sections 2 and 7 of a paper of Avila-Goüezel-Yoccoz.
For this sake, we organize this short series of posts as follows. In next section, we introduce a prototypical class of semiflows exhibiting exponential mixing. After that, we state the main exponential mixing result of this post (an analog of Theorem 7.3 of Avila-Goüezel-Yoccoz paper) for such semiflows, and we reduce the proof of this mixing property to a Dolgopyat-like estimate on weighted transfer operators. Finally, the next post of the series will be entirely dedicated to sketch the proof of the Dolgopyat-like estimate.
1. Expanding semiflows
Recall that a suspension flow is a semiflow , , associated to a base dynamics (discrete-time dynamical system) and a roof function in the following way. We consider where is the equivalence relation induced by , and we let be the semiflow on induced by
Geometrically, , flows up the point , , linearly (by translation) in the fiber until it hits the “roof” (the graph of ) at the point . At this moment, one is sent back (by the equivalence relation ) to the basis at the point , and the semiflow restarts again.
In this post, we want to study the decay of correlations of expanding semiflows, that is, a suspension flow so that the base dynamics is an uniformly expanding Markov map and the roof function is a good roof function with exponential tails in the following sense.
Remark 1 Avila-Gouëzel-Yoccoz work in greater generality than the setting of this post: in fact, they allow to be a John domain and they prove results for excellent hyperbolic semiflows (which are more common in “nature”), but we will always take and we will study exclusively expanding semiflows (which are obtained from excellent hyperbolic semiflows by taking the “quotient along stable manifolds”) in order to simplify our exposition.
Definition 1 Let , be the Lebesgue measure on , and be a finite or countable partition of modulo zero into open subintervals. We say that is an uniformly expanding Markov map if
- is a Markov partition: for each , the restriction of to is a -diffeomorphism between and ;
- is expanding: there exist a constant and, for each , a constant such that for each ;
- has bounded distortion: denoting by the inverse of the Jacobian of and by the set of inverse branches of , we require that is a function on each and there exists a constant such that
for all and . (This condition is also called Renyi condition in the literature.)
Remark 2 Araújo and Melbourne showed recently that, for the purposes of discussing exponential mixing properties (for excellent hyperbolic semiflows with one-dimensional unstable subbundles), the bounded distortion (Renyi condition) can be relaxed: indeed, it suffices to require that is a Hölder function such that the Hölder constant of is uniformly bounded for all .
Example 1 Let be the finite partition (mod. ) of provided by the two subintervals , . The map given by for is an uniformly expanding Markov map (preserving the Lebesgue measure ).
An uniformly expanding map preserves an unique probability measure which is absolutely continuous with respect to the Lebesgue measure . Moreover, the density is a function whose values are bounded away from and , and is ergodic and mixing.
Indeed, the proof of these facts can be found in Aaronson’s book and it involves the study of the spectral properties of the so-called transfer (Ruelle-Perron-Frobenius) operator
(as it was discussed in Theorem 1 of Section 1 of this post in the case of a finite Markov partition , )
Definition 2 Let be an uniformly expanding Markov map. A function is a good roof function if
- there exists a constant such that for all ;
- there exists a constant such that for all and all inverse branch of ;
- is not a –coboundary: it is not possible to write where is constant on each and is .
Remark 3 Intuitively, the condition that is not a coboundary says that it is not possible to change variables to make the roof function into a piecewise constant function. Here, the main point is that we have to avoid suspension flows with piecewise constant roof functions (possibly after conjugation) in order to have a chance to obtain nice mixing properties (see this post for more comments).
Definition 3 A good roof function has exponential tails if there exists such that .
The suspension flow associated to an uniformly expanding Markov map and a good roof function with exponential tails preserves the probability measure
on . Note that is absolutely continuous with respect to (because is absolutely continuous with respect to ).
Remark 4 All integrals in this post are always taken with respect to or unless otherwise specified.
Remark 5 In the sequel, AGY stands for Avila-Gouëzel-Yoccoz.
2. Statement of the exponential mixing result
Let be an expanding semiflow.
for all and for all .
Remark 6 By applying this theorem with in the place of , we obtain the classical exponential mixing statement:
Remark 7 This theorem is exactly Theorem 7.3 in AGY paper except that they work with observables and belonging to Banach spaces and which are slightly more general than (in the sense that ). In fact, AGY need to deal with these Banach spaces because they use their Theorem 7.3 to deduce a more general result of exponential mixing for excellent hyperbolic semiflows (see their paper for more explanations), but we will not discuss this point here.
The remainder of this post is dedicated to the proof of Theorem 4.
2.1. Reduction of Theorem 4 to Paley-Wiener theorem
For this sake, we will use the following classical theorem in Harmonic Analysis (stated as Theorem 7.23 in AGY paper):
Theorem 5 (Paley-Wiener) Let be a bounded measurable function and denote by (defined for with ) the Laplace transform of .Suppose that can be analytically extended to a function defined on a strip in such a way that
Then, there exists a constant and a full measure subset such that
for all .
In other words, the Paley-Wiener theorem says that a bounded measurable function decays exponentially whenever its Laplace transform admits a nice analytic extension to a vertical strip to the left of the imaginary axis .
In our context, we will produce such an analytic extension by writing the Laplace transform of the correlation function as an appropriate geometric series of terms depending on . In fact, the th term of this series will have a clear dynamical meaning: it will be related to the pieces of orbit hitting times the graph of the roof function.
More concretely, for each , we decompose the phase space as
- is the subset of points whose piece of orbit hits the graph of the roof function at least once,
- and is the subset of points whose piece of orbit does not hit the graph of the roof function.
Let us denote by
the corresponding decomposition of the correlation function (3).
The fact that the roof function has exponential tails implies that the probability of the event that the piece of orbit does not hit the graph of becomes exponentially small for large.
Thus, it is not surprising that the following calculation shows that decays exponentially fast as grows:
where (by the exponential tails condition on ).
In particular, the proof of Theorem 4 is reduced to show that decays exponentially. As we already mentioned, the basic idea to achieve this goal is to use the Paley-Wiener theorem and, for this reason, we want to write the Laplace transform as an appropriate series by decomposing the phase space accordingly to the number of times that a certain piece of orbit hits the graph of the roof function:
where is defined by (1). Note that this is valid for any with .
An economical way of writing down this series uses the partial Laplace transform of a function defined by the formula for . In this language, the previous identity gives that
Remark 8 In general, we expect the terms of the series (4) to decay exponentially because is expanding and is “dual” to the (Koopman-von Neumann) operator defined by composing functions with .
This formula (stated as Lemma 7.17 in AGY paper) is the starting point for the application of Paley-Wiener theorem to . More precisely, we can exploit it to analytically extend into three steps:
- (a) Lemma 7.21 in AGY paper: can be extended to a neighborhood of any ;
- (b) Lemma 7.22 in AGY paper: can be extended to a neighborhood of the origin ;
- (c) Corollary 7.20 in AGY paper: can be extended to a strip for some small and large in such a way that the extension satisfies (for some constant depending on and ).
Geometrically, the first two steps permit to extend to rectangles of the form
for . Indeed, this is a consequence of (a), (b) and the compactness of the segment .
Of course, these steps are not sufficient to extend to a whole strip to the left of the imaginary axis (because there is nothing preventing as ), and this is why the third step is crucial.
In summary, the combination of the three steps (a), (b) and (c) (and the fact that the function is integrable) says that has an analytic extension to a strip satisfying the hypothesis of Paley-Wiener theorem 5.
Therefore, we have that decays exponentially (and, a fortiori, the correlation function also decays exponentially) if we can establish (a), (b) and (c).
2.2. Implementation of (a), (b) and part of (c)
Observe that the series in (4) is bounded by
This hints that we should first compare the sizes of and to the sizes of and before trying to show the geometric convergence of this series (for certain values of ). In this direction, we have the following pointwise bound (compare with the equation (7.66) in AGY paper):
for any function and for all with .
Proof: Recall that . Hence, the desired estimate is trivial for . The remaining case is dealt with by integration by parts. Indeed, we have
Since the right-hand side has boundary terms bounded by and an integral term bounded by
(as ), we see that the proof of the lemma is complete.
This pointwise bound permits to control (cf. Lemma 7.18 in AGY paper):
for all with .
Proof: This is an immediate consequence of Lemma 6 and the fact that the function is integrable (by the exponential tails condition on ).
On the other hand, this pointwise bound is not adequate to control in terms of a geometric series. In fact, it is well-known that the weighted transfer operators only exhibit some contraction property when one also works with stronger norms than . In our current setting, it might be tempting to try to control the norm of in terms of the norm of . As we are going to see now, this does not quite work for directly (as the pointwise bound in Lemma 6 also involves the function which might be unbounded), but it does work for (compare with Lemma 7.18 of AGY paper):
for all with .
Proof: Recall that .
It follows from Lemma 6 that
where denotes the constant function of value one. Since it is not hard to check that the operator acting on the space of functions is bounded for , the first item of the lemma is proved.
Let us now prove the second item of the lemma. For this sake, we write . In particular, the derivative has four terms: one can differentiate the term or or the limit of integration or (resp.). Let us denote by , , and (resp.) the terms obtained in this way.
The fourth term is bounded by
Similarly, the bounded distortion property for implies that
Finally, since (by definition of good roof function) and , we see that the third term is bounded by
and the first term is bounded by
(thanks to the estimate from the first item.)
This completes the proof of the lemma.
and this is exactly the statement of Lemma 7.18 in AGY paper. This norm will show up again in the statement of the Dolgopyat-like estimate.
Once we have Corollary 7 and Lemma 8, we are ready use the estimate (6) and some classical properties (namely, Lasota-Yorke inequality and weak mixing for ) in order to implement the step (a) of the “Paley-Wiener strategy”:
Proof: It is well-known (cf. Lemma 7.8 in AGY paper) that the weighted transfer operator acting on satisfies a Lasota-Yorke inequality. We will come back to this point in the next post of this series, but for now let us just mention a key spectral consequence of a Lasota-Yorke inequality for .
By Hennion’s theorem (cf. Baladi’s book), a Lasota-Yorke inequality for implies that its essential spectral radius is and its spectral radius . In concrete terms, this means that there exists a constant such that the spectrum of is entirely contained in the ball except for possibly finitely many eigenvalues (counted with multiplicity) located in the annulus .
In other words, if one can show that has no eigenvalues of modulus , then the previous description of the spectrum of gives that for some constants and , and for all . Of course, since is a small analytic perturbation of for any in a small disk centered at , this implies that
defines an analytic extension to of .
In summary, we have reduced the proof of the lemma to the verification of the fact that has no eigenvalues of modulus (when ). As it turns out, this is an easy consequence of the spectral characterization of the weak-mixing property for the expanding semiflow : indeed, this property says that the Koopman-von Neumann operator given by composition with has no eigenvalues of modulus , and it is not difficult to see that this means that has no eigenvalues of modulus .
This proves the lemma.
Remark 10 A more direct proof of this lemma (without relying on the weak-mixing property for ) can be found in Lemma 7.21 of AGY paper.
Next, let us adapt the argument above to perform the step (b) of the Paley-Wiener strategy:
Proof: The transfer operator has a simple eigenvalue (cf. Aaronson’s book). In particular, the argument used to prove the previous lemma does not work (i.e., it is simply false that decays exponentially as ).
Nevertheless, we can overcome this difficulty as follows. For in a small open disk centered at , is an analytic perturbation of . Thus, has an eigenvalue close to , and we can write
where is the spectral projection to the eigenspace generated by the normalized eigenfunction (with ) associated to , and . Furthermore, the spectral properties of mentionned during the proof of Lemma 9 also tell us that there exist uniform constants and such that for all and .
In other terms, after we remove from the component associated to the eigenvalue , we obtain an operator with nice contraction properties.
(Here, we used that , , and )
Note that this is a completely obvious task for because , i.e., the analytic function
has a pole at .
Fortunately, the order of pole at of this function can be shown to be one by the following calculation. Since is an analytic perturbation of , we have that and . In particular,
where is the absolutely continuous invariant probability measure of . This means that has derivative at , and, a fortiori, the pole of at has order one.
has a zero at . This last fact is not hard to check: by definition, is a constant multiple of the function and , so that
and our assumption (2) was precisely that .
This proves the lemma.
Closing this post, let us reduce the step (c) to the following Dolgopyat-like estimate (compare with Proposition 7.7 in AGY paper):
for all with , , . (Here, is the norm introduced in Remark 9.)
The proof of this proposition will occupy the next post of this series. For now, let us implement the step (c) of the Paley-Wiener strategy assuming Proposition 11.
We want to use the formula (4) to define a suitable analytic extension
of to a strip of the form .
for all with and .
This proves that is an analytic extension of to such that , which are exactly the properties required in the step (c).