Yuri Lima and I have just uploaded to ArXiv our paper “Symbolic dynamics for non-uniformly hyperbolic maps with discontinuities”. The main motivation for our paper is the question of extending the celebrated (Brin prize) work of Sarig on symbolic models/Markov partitions for smooth surface diffeomorphisms to the context of billiard maps: indeed, the main result of our paper is a partial solution to a problem appearing in page 346 of Sarig’s article.
An interesting corollary of our results is a refinement of a theorem of Chernov on the number of periodic points of certain billiard maps. More precisely, for a certain class of billiard maps , Chernov proved that
where is the set of periodic points of with period and is the Kolmogorov-Sinai entropy of the Liouville measure. From our main results, Yuri and I can show that the billiard maps studied by Chernov actually satisfy:
Remark 1 Our improvement of Chernov’s theorem is “similar in spirit” to Sarig’s improvement of Katok’s theorem on the number of periodic points for smooth surface diffeomorphisms: see Theorem 1.1 in Sarig’s paper for more details.
Below the fold, we give a slightly simplified version of the main result in our paper and we explain some steps of its proof.
1. Symbolic models for certain billiard maps
Consider a planar billiard map , , where is a compact billiard table whose boundary is a finite union of smooth curves: by definition, whenever the straight line starting from in direction hits at with angle of incidence ( angle of reflection) .
Recall that a billiard map preserves the Liouville measure .
In 1986, Katok and Strelcyn showed that the so-called Pesin theory of smooth non-uniformly hyperbolic diffeomorphisms could be extended to non-uniformly hyperbolic billiard maps under mild conditions.
More concretely, a billiard map usually exhibits a singular set (related to discontinuities of , grazing collisions, etc.) and, roughly speaking, Katok and Strelcyn results say that if has reasonable geometry (e.g., the Liouville -measure of -neighborhoods of decay polynomially fast with ), then Pesin theory applies to a non-uniformly hyperbolic billiard map whose first two derivative explode at most polynomially fast as one approaches .
Philosophically speaking, the basic idea behind Katok-Strelcyn theorems is that the good exponential behavior provided by non-uniform hyperbolicity is strong enough to overcome the bad polynomial behavior near the singular set . (Of course, this is easier said than done: Katok-Strelcyn’s work is extremely technical at some places.)
In our paper, Yuri and I show that Katok-Strelcyn philosophy can also be used to extend Sarig’s theory to billiard maps:
Theorem 1 Let be any billiard map within the framework of Katok-Strelcyn’s theory (e.g., Sinai billiards, Bunimovich stadia, etc.). Then, there exists a topological Markov shift (of countable type) and a Hölder continuous map such that
- the shift codes the dynamics of , i.e., ;
- most -orbits are captured by the coding, i.e., the set has full Liouville -measure;
- is finite-to-one (and, hence, the Liouville measure on can be lifted to without increasing the entropy).
Remark 2 The main result of our paper (Theorem 1.3) deals with a more general class of surface maps with discontinuities, but its precise statement is somwhat technical: we refer the curious reader to the original article for more details.
2. Sarig’s theory of symbolic models
The general strategy to prove Theorem 1 follows closely Sarig’s methods. More precisely, given a billiard map such as a Sinai or Bunimovich billiard, we fix such that the Lyapunov exponents of with respect to the Liouville measure do not belong to the interval .
By Oseledets theorem, there is a set of full -measure such that any has the following properties:
- for all , there are unit vectors with for ;
- and ;
- the angle between and decays subexponentially:
Furthermore, the assumption that the singular set has a reasonable geometry (e.g., the logarithm of the distance to is -integrable) says that the subset consisting of points whose -orbits do not approach exponentially fast, i.e.,
also has full -measure.
One of the basic strategies to code (a full measure subset of) relies on the so-called shadowing lemma: very roughly speaking, for sufficiently small, we want the -orbit of -almost every to be shadowed by (“fellow travel with”) finitely many –generalized pseudo-orbits.
The notion of -generalized pseudo-orbits is the same from Sarig’s work: in particular, they are not defined in terms of sequences of points chosen from a countable dense subset with the property that for all , but rather in terms of sequences of double Pesin charts (taken from a countable “dense” subset ) with the property that –overlaps for all .
Here, the advantage in replacing points by Pesin charts comes from the fact that looks like a uniformly hyperbolic linear map, so that we can hope to apply the usual tools from the theory of hyperbolic systems (stable manifolds, etc.) to establish the desired shadowing lemma.
After this succint explanation of Sarig’s method for the construction of symbolic models for non-uniformly hyperbolic systems, let us now discuss in more details the implementation of Sarig’s ideas.
2.1. Linear Pesin theory
Before trying to render into an almost linear hyperbolic map in adequate (Pesin) charts, let us convert the derivative of at into a uniformly hyperbolic matrix. For this sake, we use an old trick in Dynamical Systems, namely, we introduce the hyperbolicity parameters
and angle between and . Note that and are well-defined (i.e., the corresponding series are convergent) because .
In terms of these parameters, we can define the linear map via
where is the canonical basis of .
A straightforward computation reveals that becomes a uniformly hyperbolic matrix when viewed through the linear maps , i.e.,
where and .
Of course, the conversion of the non-uniformly hyperbolic map into a uniformly hyperbolic matrix has a price: while the norm of is , a simple calculation shows that the Frobenius norm of its inverse is
In particular, “explodes” when the hyperbolicity parameters degenerate (e.g., approaches zero).
2.2. Non-linear Pesin theory
After converting into a uniformly hyperbolic matrix via , we want to convert into an almost (uniformly hyperbolic) linear map near . For this sake, we compose with the exponential map to obtain the Pesin chart
In this way, is a map fixing such that
where and .
Of course, this means that is an almost (hyperbolic) linear map in some neighborhood of , but this qualitative information is not useful: we need to control the size of this neighborhood of (in order to ensure that a countable set of [double] Pesin charts suffice to code the dynamics of on a full -measure set of points of ).
In this direction, we introduce a small parameter depending on , and the distance of to (whose precise definition can be found at page 10 in our paper). Then, a simple calculation (cf. Theorem 3.3 in our paper) shows that, for all in the square , one has
where and are smooth functions whose -norms on are smaller than .
In fact, our choice of involves and in order to control the distortion create by the linear maps and in the definition of .
On the other hand, the dependence of on is a novelty with respect to Sarig’s paper and it serves to control the eventual polynomial explosion of the first two derivatives and of near (i.e., and for some ).
Once we dispose of good formulas for on the Pesin charts of and , we want to “discretize” the set of Pesin charts: since our final goal is to code most -orbits with a countable set of Pesin charts, we do not want to keep all , .
Here, the basic idea is that we can safely replace by whenever has (essentially) the same features of , i.e., it is an almost (hyperbolic) linear map on the square .
Since is defined in terms of , it is not surprising that and and, a fortiori, and are close whenever the points and are close and the matrices and are close.
This motivates the definition of -overlap of two Pesin charts.
Definition 2 Given and , denote by the restriction of to the square . We say that -overlaps if and
As the reader might suspect, this definition is designed so that if -overlaps , then the hyperbolicity parameters , , of and are close, and is –-close to the identity (on a square for some ): see Proposition 3.4 in our paper.
By exploiting this information, we show (in Theorem 3.5 of our paper) that if -overlaps , then
where and are smooth functions whose -norms on are smaller than .
2.3. Generalized pseudo-orbits
The graph associated to the topological Markov shift coding will be defined in terms of two pieces of data: its vertices are –double charts and its edges connect a double chart whose “iterate” under has -overlap with another double chart.
Definition 3 A -double chart is a pair of Pesin charts whose parameters belong to the countable set .
Remark 3 The philosophy in the consideration of and is that, contrary to the uniformly hyperbolic case, the forward and backward behavior of non-uniformly hyperbolic systems might be very different, hence we need to control them separately.
- (GPO1) -overlaps and -overlaps .
- (GPO2) and .
Remark 4 GPO stands for “generalized pseudo-orbit”. The second condition (GPO2) is a greedy way of ensuring that the parameters and (controlling , and, thus, the hyperbolicity parameters , ) are the largest possible.
Definition 5 A -generalized pseudo-orbit is a sequence of -double charts such that we have an edge for all .
The fact that -generalized pseudo-orbits are useful for our purposes is explained by the following result (cf. Lemma 4.6 in our paper):
Lemma 6 Every -generalized pseudo-orbit shadows an unique point, i.e., there exists an unique such that
for all .
The proof of this shadowing lemma follows the usual ideas in Dynamical Systems: first, one defines stable/unstable manifolds using the Hadamard-Perron graph transform method, and, secondly, one shows that the unique point shadowed by is precisely the unique intersection point between the stable and unstable manifolds. In particular, we use here that the fast (exponential) pace of the dynamics along “almost stable/unstable manifolds” (called -admissible manifolds) is sufficiently strong to apply Sarig’s arguments even if and are allowed to explode at a slow (polynomial) pace near the singular set .
2.4. Coarse graining
The next step is to select a countable collection of -double charts such that the corresponding -generalized pseudo-orbits shadow a set of full -measure.
- is discrete: for all , the set is finite;
- is sufficient to code most -orbits: there exists of full -measure so that if , then there exists a -generalized pseudo-orbit shadowing ;
- all elements of are relevant for the coding: given , there exists a -generalized pseudo-orbit with that shadows a point in .
In a nutshell, the proof of this theorem is a pre-compactness argument. More precisely, for each in an appropriate subset (of full -measure), we consider the parameters
controlling the Pesin charts , , . Since the spaces , and are pre-compact (or, more precisely, for all , the sets , and are compact), we can select a countable subset of which is dense in the following sense: for all and , there exists such that
and, for each ,
In terms of , the countable collection of -double charts verifying the conclusions of the theorem is essentially .
This theorem yields a topological Markov shift associated to the graph whose set of vertices is and whose edges are (cf. Definition 4), i.e., is the set of bi-infinite (-indexed) paths on and is the shift dynamics on . Since any is a -generalized pseudo-orbit, we have a map , where
is the point shadowed by .
The map has the following properties (cf. Proposition 5.3 in our paper):
Proposition 8 Every has finite valency in (and, hence, is locally compact). Moreover, is Hölder continuous, and codes most -orbits (i.e., and has full -measure).
The first part of this proposition follows from the discreteness of (cf. Theorem 7), the Hölder continuity of is a consequence of the nice dynamical properties of almost stable/unstable manifolds, and the fact that codes most -orbits (i.e., has full Liouville measure) is deduced from the second item of Theorem 7.
2.5. Inverse theorem
In general, is not finite-to-one, i.e., might not satisfy the last conclusion of Theorem 1. Therefore, we need to refine before trying to use to induce a locally finite cover of a subset of of full -measure.
For this sake, it is desirable to understand how loses injectivity, and, as it turns out, this is the content of the so-called inverse theorem (cf. Theorem 6.1 in our paper):
then all relevant parameters (distance, angle, hyperbolicity, etc.) are close together:
- for all ;
- and for all ;
- for all ;
- for all ;
- for all ;
- for all , is –-close to (for an adequate choice of ) on the square .
Intuitively, this theorem says that “tends” to be finite-to-one because the parameters of (a -recurrent) “essentially” determine the parameters of any (-recurrent) with , so that the discreteness of (cf. Theorem 7) implies that there are not many choices for such .
The proof of the inverse theorem is the core part of both Sarig’s paper and our work. Unfortunately, the explanation of its proof is beyond the scope of this post (because it is extremely technical), and we will content ourselves in pointing out that the presence of the singular set introduces extra difficulties when trying to run Sarig’s arguments: for example, contrary to Sarig’s case, the parameter also depends on , so that we need to take extra care in the discussion of the fourth item of the inverse theorem above.
2.6. Bowen-Sinai refinement method
Once we dispose of the inverse theorem in our toolkit, the so-called Bowen-Sinai refinement method (for the construction of Markov partitions) explained in Sections 11 and 12 of Sarig’s paper can be used in our context of billiard maps without any extra difficulty: see Section 7 of our paper for more details.
For the sake of convenience of the reader, let us briefly recall how Bowen-Sinai method works to convert the coding into the desired coding satisfying the conclusions of Theorem 1.
First, we start with the collection , where
The s/u-fiber of is
where is any -recurrent element of with . (The nice properties of “almost stable/unstable manifolds” ensure that is well-defined [i.e., it doesn’t depend on the particular choice of ].)
It is not difficult to show that is a cover of a full -measure subset which is locally finite (i.e., for all , the set is finite). Moreover, has local product structure (i.e., for all , , ), and is a Markov cover (i.e., for any -recurrent with , one has and ).
Now, we refine according to the ideas of Bowen and Sinai. More concretely, we take the Markov cover and, for any , we consider:
Then, we define as the partition induced by the collection
At this point, we can complete the proof of Theorem 1 by proving that is a countable Markov partition such that the graph with set of vertices and edges whenever induces a topological Markov shift
with the desired properties, namely, is Hölder continuous, finite-to-one, and .