The set of points with Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms

Abstract The papers [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc. 26(2) (2013), 341–426] and [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn. 13 (2018), 43–113] constructed symbolic dynamics for the restriction of 
$C^r$
 diffeomorphisms to a set 
$M'$
 with full measure for all sufficiently hyperbolic ergodic invariant probability measures, but the set 
$M'$
 was not identified there. We improve the construction in a way that enables 
$M'$
 to be identified explicitly. One application is the coding of infinite conservative measures on the homoclinic classes of Rodriguez-Hertz et al. [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys. 306(1) (2011), 35–49].

This paper studies the symbolic dynamics of non-uniformly hyperbolic diffeomorphisms f : M → M. In this case, the papers [BO18,Sar13] constructed a countable Markov partition for a subset M ⊆ M of the manifold, which carries all 'sufficiently hyperbolic' ergodic invariant probability measures (a precise statement is given in §2). See [Lim20,LM18,LS19] for other coding results in the non-uniformly hyperbolic setup and [Lim19] for a survey of recent advances in the construction of symbolic dynamics for non-uniformly hyperbolic systems.
Although M is 'big' from the point of view of hyperbolic ergodic invariant probability measures, it is usually not equal to the entire manifold. For example, it does not contain elliptic islands, or orbits with zero Lyapunov exponents.

Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms 3245
The papers [BO18,Sar13] do not identify M explicitly. In this paper we modify the construction of the Markov partition done there to yield a countable Markov partition for which the set M can be determined explicitly.
Our main result says that M equals the set of orbits which are summable and recurrently weakly temperable.
Summability and recurrent weak temperability are defined in §2.2. The precise statement, Theorem 4.5, is given in §4. Finally, §5 adopts the results of [BCS] on transitive codings of homoclinic classes for our setup.
An important motivation for this work is an ongoing project on generalized SRB measures. These are possibly infinite invariant measures with strong 'physical' properties.
To study them symbolically, we need to know if they are carried by the set M . This does not follow from the results of [BO18,Sar13], which only identify M modulo hyperbolic probability measures.
This paper treats diffeomorphisms. The case of flows brings in new difficulties, because of issues related to singularities in the Poincaré section. The paper [LS19] codes a smaller set than the set of Lyapunov regular points. The author has been informed by Lima that together with Buzzi and Crovisier they now have a coding which captures a larger set than the set of Lyapunov regular orbits [Lim].

Statement of results
Let M be a compact Riemannian manifold without boundary, of dimension d ≥ 2. Let f ∈ Diff 1+β (M), β > 0 (i.e. f is invertible, f , f −1 are differentiable, and both the derivatives d x f , d x f −1 are β-Hölder continuous functions of x).

General notation.
(1) For every a, b ∈ R, c ∈ R + , a = e ±c · b means e −c · b ≤ a ≤ e c · b, and a ∧ b := min{a, b}.
(2) For all x ∈ M, ·, · x : T x M × T x M → R is the inner product on the tangent space of x given by the Riemannian metric. | · | x : T x M → R is the norm induced by the inner product, |ξ | 2 x := ξ , ξ x , for all ξ ∈ T x M. We often omit the x subscript of the inner product and of the norm, when the tangent space in domain is clear from the context. This work uses tools which were previously developed in [BO18,Sar13]. In the following subsection, we introduce two notions of hyperbolic points, in order to have a canonical (in this context, 'canonical' means definitions which do not rely on a specific construction of symbolic dynamics, but which depend only on the quality of hyperbolicity of the orbit of the point) characterization for a set of points which our symbolic extension codes (see [BO18,Sar13]).

S. Ben Ovadia
(2) χ-hyperbolicity. The set of χ-hyperbolic points is may not be unique. However, for -weakly temperable points and small enough > 0 (see Definition 2.3), the decomposition is unique and invariant. See remark (3) in §2.4 for details.
where ·, · 2 is the Euclidean inner product on R d . In addition, where D s (x), D u (x) are square matrices of dimensions s(x), u(x) respectively, and The map C χ (x) is called the Lyapunov change of coordinates, and this theorem is a version of the Pesin-Oseledec reduction theorem [KM95,Pes76], which we prove in [ (2) for all n ∈ Z, q(f n (x)) ≤ Q (f n (x)).

Symbolic dynamics.
THEOREM 2.4. For all χ > 0 such that there exists a periodic hyperbolic point p ∈ χ-hyp, there exists χ > 0 (which only depends on M, f , β, χ) such that for all 0 < ≤ χ there exists a countable and locally finite directed (2) π is a Hölder continuous map with respect to the metric d(u, v) := exp(− min{i ≥ 0 :

S. Ben Ovadia
This theorem is the content of [Sar13, Theorem 4.16] in dimension 2, and [BO18, Theorem 3.13] in any dimension. V is a collection of double Pesin charts (see Definition 3.3), which is discrete (every v ∈ V is a double Pesin chart of the form v = ψ p s ,p u x with 0 < p s , p u ≤ Q (x); and discreteness means that for all η > 0 : #{v ∈ V : v = ψ p s ,p u x p s ∧ p u > η} < ∞).

Remark.
(1) In Definition 2.2, the 48/β exponent is an artifact of our proof, and any sufficiently large power of C −1 χ (x) in the definition of Q (·) suffices. Altering the power changes both the set of coded points, and the coding (see §3.1).
(3) Q (·) depends only on and on C −1 χ (·) , which depends only on the decomposition T · M = H s (·) ⊕ H u (·). By equation (1), if x ∈ χ-summ is also -weakly temperable (and is small with respect to χ, β, as imposed by the assumption ≤ χ from Theorem 2.4), then the decomposition T x M = H s (x) ⊕ H u (x) must be unique. Therefore, Q (·) is defined canonically for -weakly temperable points, and does not depend on the choice of C χ (·). Thus, for all ∈ (0, χ ], WT χ and RWT χ are defined canonically. (4) RWT χ is of full measure with respect to every invariant probability measure carried by χ-summ.
Proof. We prove the statement for RWT χ ; the proof for WT χ is similar. Let x ∈ RWT χ , and let q : , for all n ∈ Z, and thus lim sup n→±∞ q (f n (x)) > 0. Since {e − /3 } ∈N is closed under multiplication, it follows that q satisfies the assumptions of recurrent -weak temperability for x, and so x ∈ RWT χ .
3. Review of [BO18,Sar13] Throughout this section, f : M → M is a C 1+β diffeomorphism of a compact Riemannian manifold without boundary M, and χ > 0.
Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms 3249 , and let C χ (x) be a Lyapunov change of coordinates as in Theorem 2.1. Define, for all ξ s ∈ H s (x), ξ u ∈ H u (x), are Pesin charts with the same map and concentric domains.

Main steps of
(1) A set of hyperbolic points. There exists a set, NUH * χ , of Lyapunov regular χ-hyperbolic points, such that for all For all x ∈ NUH * χ , and for every sequence (η n ) n∈Z such that 0 < η n ≤ Q (f n (x)), η n ∈ {e − /3 } ∈N and η n /η n+1 = e ± for all n ∈ Z, there exists a sequence (ψ η n x n ) n∈Z of elements of A such that for all n ∈ Z: for all n ∈ Z, and there exists y ∈ ] that G is locally finite: every vertex has finitely many incoming and outgoing edges. We have (6) This is the induced topological Markov shift (TMS). The local finiteness of G implies that is locally compact. admits a factor map π : → M as given by Theorem 2.4.
Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms 3251 where 0 < q ≤ Q (x), and F u is a C 1+β/3 function such that max R q (0) | F u | ∞ ≤ Q (x). An s-manifold in ψ x is defined similarly as a set (t 1 , . . . , t s(x) ), . . . , F s d (t 1 , . . . , t s(x) )) : with the same requirements for F s and q. We will use the superscript u/s in statements which apply to both the u case and the s case. The function F = F u/s is called the representing function of V u/s at ψ x . The parameters of a u/s manifold in ψ x are: Notice that the dimensions of an s-or a u-manifold in ψ x depend on x. Their sum is d.

Proof of main result
We now present the changes to the construction of χ , V , in [BO18], which will allow us to identify the coded set π [ # ]. The main modifications are to replace NUH * χ by WT χ in the coarse graining, to replace NUH * by RWT χ in the definition of relevance, and to replace χ by min{ χ , χ/2 }.
(2) Given ∈ (0, χ ], A is a discrete and sufficient set of Pesin charts as in §3.1 step (2), but given by the coarse graining process for Pesin charts with centers in WT χ , instead of NUH * χ . for all n ∈ Z, and there exists y ∈ RWT χ such that f n (y) ∈ ψ x n [R Q (x n ) (0)] for all n ∈ Z (instead of imposing y ∈ NUH * χ , as in §3.1 step (4)(a)).
is a countable locally finite directed graph (the local finiteness of G follows from the discreteness of V ; see §2.3).
(5) := {u ∈ V Z such that (u i , u i+1 ) ∈ E , for all i ∈ Z} is a locally compact TMS induced by G .

Remark. Notice that the map π :
→ M from Theorem 2.4 is well defined on . This is true since, given a chain u ∈ , π(u) is the unique intersection point of V u (u) and V s (u) (see Claim 3.7). The same construction extends to , since all arguments which lead to Claim 3.7 for the definition of V s , V u (see [BO18, Theorem 3.6], and the graph transform argument) require only χ-summability of the charts' centers for the Pesin-Oseledec reduction theorem (see Theorem 2.1).
The following lemma is an improvement of [BO18, Lemma 4.5].
LEMMA 4.2. Let v ∈ such that π(v) ∈ RWT χ . Write W s = V s (v) and z = π(v), and let r ∈ (χ/2, χ). Then there exists C v 0 ,z > 0 which depends only on v 0 , χ, f , and z such that for all y ∈ W s , Notice that the bound on the right-hand side is uniform in y and r, which is the main improvement compared to [BO18, Lemma 4.5].
Proof. Let q : {f n (z)} n∈Z → (0, ) be given by the -weak temperability of z. By definition, q (f n (z)) ≤ Q (f n (z)), for all n ∈ Z. So, by [Sar13, Lemma 4.6], there exists (p s n ) n∈Z , (p u n ) n∈Z ⊆ {e − /3 } ≥0 such that q (f n (z)) ≤ p s n ∧ p u n ≤ Q (f n (z)), for all n ∈ Z, and u := (ψ p s i ,p u i f i (z) ) i∈Z is a chain (i.e. u i → u i+1 , for all i ∈ Z, recall §3.1 step (3)). Notice that the sequence (p s n ∧ p u n ) n∈Z is -weakly temperable and recurrent (see [Sar13,Lemma 4.4]). Let y ∈ W s .
The idea of the proof follows the steps of [BO18, Lemma 4.5] for push-forwards of y, z (with r replacing χ and χ replacing χ z , and -weak temperability replacing the assumption lim n→∞ (1/n) log C −1 χ (f n (z)) = 0); and then uses a version of [BO18,Lemma 4.6] to pull back the refined bounds. The reason why this works for r ∈ (χ/2, χ), and not just for χ as required in [BO18,Lemma 4.6], is that there is no distortion of bounds by the non-complete overlap of charts in the chain u, as u is a chain over a real orbit and not merely a pseudo-orbit. Set The reason why the minimum is not over an empty set is that diam f n [W s ] (f n [W s ]) decreases exponentially faster than q (f n (z)) (by -weak temperability), and that z is χ-summable, and so r -hyperbolic for all r ∈ (r, χ). For full details, see the claim within the proof of [BO18, Lemma 4.5], with r replacing χ and χ replacing χ z ; the assumption lim n→∞ (1/n) log C −1 χ (f n (z)) = 0 is replaced by the -weak temperability of z. Let k z,r ≥ 0 such that for all k ≥ k z,r , n k ≥ n z,r . Thus, for all k ≥ k z,r , for all y ∈ e −m(2(χ −r))/3 < ∞. (2) The bound on the right-hand side depends on r and z, but does not depend on k (as long as k ≥ k z,r ). We now wish to use [BO18,Lemma 4.6] with the chain u, and gain a bound for points in W s , instead of points in f n k [W s ]. This requires justification, since in [BO18, Lemma 4.6] the exponential factor in the sum is χ, and we wish to apply it to a sum with an exponential factor of r. The proof of [BO18, Lemma 4.6] starts by breaking the quotient which we wish to estimate into two factors (see equation (14) in the proof). The first factor is the improvement in the quotient which wish to bound, due to the pull-back by f −1 . The estimates of the first factor (see equation (17) in the proof) remain unchanged (since C −1 r (f n k (z)) ≤ C −1 χ (f n k (z)) ). The second factor is due to the distortion by non-complete overlap of charts (which does not apply to our case since u is a chain over the actual orbit of z, and not merely a pseudo-orbit, hence the overlap is complete), and due to the distance in the tangent bundle between the tangent vectors π z ξ and η (in the notation of the proof of [BO18, Lemma 4.6]; see steps 1 and 2 in the proof). The assumption N = 0 (i.e. W s ⊆ V s (u)) allows the estimates of the distance in the tangent bundle to remain as in the proof.
Thus, we may apply [BO18,Lemma 4.6] to the chain u. Since the bound in equation (2) is uniform in k, we may apply [BO18, Lemma 4.6] as many times as we need, starting from u n k where k is as large as we wish. Each time we apply [BO18, Lemma 4.6], the quotient Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms 3255 which we wish to estimate either improves, or is bounded by e ± √ , and every time we hit an element of {u n j } it improves by a definite amount. Since we are free to choose k as large as we wish, eventually it will drop below e 4 √ and will stay there. It follows that Part 2. We now wish to treat the case where N > 0. Notice that, for all y ∈ W s , ξ ∈ T y W s ,  Proof. Assume by contradiction that there exist δ > 0, y ∈ W s , ξ ∈ T y W s such that ∞ m=0 |d y f m ξ | 2 e 2χm ≥ C v 0 ,z · e 4 √ +δ C −1 χ (z) 2 . Then choose N > 0 such that This is a contradiction to Lemma 4.2.
LEMMA 4.4. Let u ∈ # . Then x := π(u) is χ-summable, and Proof. This lemma is the same as Claim 2 in the proof of [BO18, Lemma 4.7], except for the fact that in [BO18,Lemma 4.7] we have # instead of # . We will now explain how to modify the argument of [BO18, Lemma 4.7] to handle # . The point is to deal with --relevance (see Definition 4.1), instead of -relevance (see §3.1).

S. Ben Ovadia
Assume without loss of generality that there exists n k ↑ ∞ such that u n k = u 0 for all k ≥ 0. By the --relevance of u 0 , there exists a chain w ∈ ∩ [u 0 ] such that z := π(w) ∈ RWT χ . This way, we may use Corollary 4.3 to replace [BO18,Lemma 4.5] in the proof of [BO18, Lemma 4.7] (i.e. to bound S(·, ·) uniformly on the unit tangent bundle of V s (w)). The statement of Claim 2 in the proof of [BO18,Lemma 4.7] gives what we wanted to show.
Step 3. In order to show recurrent -weak temperability, we wish to compare . This is similar to Claim 2 in the proof of [BO18,Lemma 4.7] and [BO18, Proposition 4.8], except that we must use # instead of # . The details are as follows.
With this is mind, [BO18, Lemma 4.7 (Claim 2)] can be carried out verbatim, and so there exists a linear invertible map π s S(x 0 , π s x 0 ξ). A similar statement holds for π u x 0 : T z V u (u) → H u (x 0 ) and U(·, ·). π s x 0 and π u x 0 extend to the invertible linear map π x 0 : T z M → T x 0 M by π x 0 | T z V s (u) = π s x 0 and π x 0 | T z V u (u) = π u x 0 . It then follows from the proof of [BO18, Proposition 4.8] that π x 0 , π −1 Thus, by Corollary 3.2, C −1 Similarly, by considering the shifted sequence, C −1 χ (f n (z)) / C −1 χ (x n ) = e ±3 √ , for all n ∈ Z.
Step 4. We can now continue to show recurrent -weak temperability. Define q : {f n (z)} n∈Z → {e − /3 } ∈N by q(f n (z)) := b · p s n ∧ p u n , where b := max{t ∈ Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms 3257 By the definition of V , p s n ∧ p u n ∈ {e − /3 } ∈N , for all n ∈ Z; thus, since {e − /3 } ∈N is closed under multiplication, the definition of q is proper. q satisfies the assumptions of recurrent -weak temperability as follows.
(3) Symbolic local product structure. For all R ∈ R , for all x, y ∈ R, there exists z := [x, y] R ∈ R, such that for all i ≥ 0, (3) Every pair of partition members R, S ∈ R is said to be -affiliated if there exists Remark. By Corollary 4.7 and Theorem 4.8, every partition member of R has only a finite number of partition members -affiliated to it.
THEOREM 4.10. Given from Definition 4.9, there exists a factor map π : → M such that the following statements hold.
(⊆). Let R, S ∈ R such that there exists x ∈ R ∩ f −1 [S]. Let u ∈ V such that R ⊆ Z (u). Then there exists u ∈ # ∩ [u] such that π(u) = x, and so S ⊆ Z (u 1 ). Given Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms 3259 a chain R ∈ # , choose Z (u 0 ) ⊇ R 0 , and construct this way a chain u ∈ such that R i ⊆ Z (u i ) for all i ∈ Z. By the uniqueness of a shadowed orbit, π(u) = π(R). By Corollary 4.7, and the pigeonhole principle, u ∈ # .

Ergodic homoclinic classes, maximal irreducible components and coding infinite conservative measures
The aim of this section is to construct an irreducible coding of an ergodic homoclinic class, which lifts all conservative (possibly infinite) measures on it (see definition below), as an extension to the preceding result in [BCS], which only treats probability measures.
In this section is fixed and equals χ . The subscript on , # , R , , # , V , Z will be omitted to ease notation.
Let p be a periodic point in χ-summ. Since p is periodic, C −1 χ (·) is bounded along the orbit of p, and therefore p ∈ RWT χ . Every point x ∈ RWT χ is (recurrently) codable, and so has a local stable manifold V s (x) (e.g. V s (u), u ∈ π −1 [{x}] ∩ # ), and a global stable manifold W s (x) := n≥0 f −n [V s (f n (x))] (similarly for a global unstable manifold). where denotes transverse intersections of full codimension, o(p) is the (finite) orbit of p, and W s (·), W u (·) are the global stable and unstable manifolds of a point (or points in an orbit), respectively.
This notion was introduced in [RHRHTU11], with a set of Lyapunov regular points replacing RWT χ . Every ergodic conservative χ-hyperbolic measure is carried by a χ-ergodic homoclinic class of some periodic hyperbolic point.

Remark.
Notice that H χ (p) ⊂ W s (o(p)) W u (o(p)) ≡ Newhouse's definition of a homoclinic class [New72]; in particular, H χ (p) is not necessarily closed. (1) Define the relation ∼⊆ R × R by R ∼ S ⇐⇒ there exist n RS , n SR ∈ N such that R such that R 0 = R, R n RS = S, R n SR +n RS = R and f −1 [R i ] ∩ R i−1 = ∅ for all 1 ≤ i ≤ n RS + n SR . The relation ∼ is transitive and symmetric. When restricted to {R ∈ R : R ∼ R}, it is also reflexive, and thus an equivalence relation. Denote the corresponding equivalence class of some representative R ∈ R, R ∼ R by R .
(2) A maximal irreducible component in , corresponding to R ∈ R such that R ∼ R, is LEMMA 5.4. Let p ∈ χ-summ such that there exists l ∈ N such that if f l (p) = p, then p ∈ χ-hyp.
Proof. We prove an exponential contraction strictly stronger than e −χ on H s (p). The case for H u (p) is similar. First assume that f (p) = p. Since p is χ-summable, for all ξ ∈ H s (p) with |ξ | = 1, ∞ m=0 |d p f m ξ | 2 e 2χm < ∞. Let {ξ i } s(p) i=1 be an orthonormal basis for H s (p) (with respect to ·, · p , the Riemannian form at