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.