Symbolic dynamics for non uniformly hyperbolic diffeomorphisms of compact smooth manifolds

We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of O. Sarig for surfaces. These partitions allow us to obtain symbolic coding on invariant sets of full measure for all hyperbolic measures whose Lyapunov exponents are bounded away from zero by a constant. Applications include counting results for hyperbolic periodic orbits, and structure of hyperbolic measures of maximal entropy.

0. Introduction 0.1.Main results.Let G be a directed graph with a countable collection of vertices V s.t.every vertex has at least one edge coming in, and at least one edge coming out.The topological Markov shift (TMS) associated to G is the set equipped with the left-shift σ : Σ → Σ, σ((v i ) i∈Z ) = (v i+1 ) i∈Z , and the metric d(u, v) := exp(− min{n ∈ N 0 : u n = v n or u −n = v −n }).Thus Σ is a complete separable metric space.Σ is compact iff G is finite.Σ is locally compact iff every vertex of G has finite degree.A subshift of the TMS is a subset of Σ which corresponds to a subgraph of G with the same properties, the same metric and equipped with the left-shift.
For a given TMS Σ, we define: Notice that by the Poincaré recurrence theorem every σ-invariant probability measure gives Σ # full measure.Furthermore, notice that every periodic point of σ is in Σ # .
Let f be a C 1+β diffeomorphism on a compact smooth boundary-less manifold M of a dimension greater than 1.Here b ∈ (0, 1) is the Hölder exponent of df .
We say that an ergodic f -invariant probability measure µ is a hyperbolic measure, if it has no zero Lyapunov exponents, and there are at least one positive and one negative exponent.For such a hyperbolic measure we define χ(µ) := min{|χ i | : χ i is a Lyapunov exponent of µ}, and if 0 < χ ≤ χ(µ), we say µ is χ-hyperbolic.Similarly, we call a Lyapunon regular point in the manifold χ-hyperbolic, if its Lyapunov exponents are bounded away from 0 by χ.
Remark: Our bound for the Hölder exponent of π χ decays to zero as χ → 0 (see the proof of proposition 1.3.20 and theorem 1.3.21).We denote the set of states of Σ χ by V χ .
Theorem 0.1.2.For every χ-hyperbolic measure µ on M there is an ergodic σ-invariant probability measure μ on Σ χ , such that µ equals μ • π −1 χ and they both have the same metric entropy.Remark: The correspondence of measures in theorem 0.1.2is in-fact two-sided, we explain it briefly: if μ is an ergodic σ-invariant probability measure on Σ χ , then µ := μ • π −1 χ is an ergodic f -invariant probability measure on M , and µ has the same entropy as μ because π χ is finite-to-one on Σ # χ , which is of full μ-measure as argued.π χ is into the set of χ-hyperbolic points, hence µ would be χ-hyperbolic as well.
Theorem 0.1.3.For Σ χ from theorem 0.1.1,there exists a function ϕ χ : V χ × V χ → N s.t. for every x ∈ M which can be written as x = π χ ((v i ) i∈Z ) with v i = u for infinitely many negative i, and v i = w for infinitely many positive i: |π −1 χ [{x}]| ≤ ϕ χ (u, w).0.2.Applications.Theorem 0.2.1.Suppose f is a C 1+β diffeomorphism of a compact smooth boundary-less manifold of a dimension greater than 1, and suppose f has a hyperbolic measure which realizes the topological entropy.Then ∃p ∈ Ns.t.lim inf n→∞,p|n e −nhtop(f ) P n (f ) > 0 where P n (f ) := #{x ∈ M : f n (x) = x} and h top (f ) is the topological entropy.
Theorem 0.2.2.Suppose f is a C 1+β diffeomorphism of a compact smooth boundary-less manifold of a dimension greater than 1.Then f possesses at most countably many hyperbolic measures which realize the topological entropy.
1.1.1.Non-uniform hyperbolicty.By the Oseledec theorem, we get the splitting of T x M , for any regular point (a point for which the Oseledec theorem holds), into subspaces H i (x) corresponding to the Lyapunov exponent χ i (x).The splitting T x M = ⊕ k(x) i=1 H i (x) can be used to diagonalize the action of df .Denote dim(H i (x)) = l i (x).From the Oseledec theorem we also get the almost-everywhere (w.r.t any invariant measure) corresponding limit for the cocycle d x f n : Λ(x) := lim n→∞ ((d x f n ) t d x f n ) 1 2n .Points for which this limit exists are called Lyapunov regular.
We will restrict ourselves for cases where s(x) ∈ {1, ..., d−1}-the hyperbolic case.The assumption of s(x) = 0, d doesn't follow from the rest of our assumptions by the example of the linear map z → 2z on the Riemann sphere, with the Dirac measure at 0. This would have only positive Lyapunov exponents, with 0 metric entropy for a smooth dynamical system.
(2) The non-uniformly hyperbolic set of points is defined for every χ > 0 as follows: Where χ i are the Lyapunov exponents, given by the Oseledec theorem.
Notice that due to the fact that χ i are f -inv.functions, χ i are const.a.e. for every ergodic invariant measure.
(3) We can thus define the following quantity for such measures which only depends on the measure (in oppose to depending on a point w.r.t to a measure): • For some point x ∈ N U H χ : H s (x) := ⊕ i:χi<0 H i (x), H u (x) := ⊕ i:χi>0 H i (x).
• For two linear vector spaces V and W , GL(V, W ) is the space of invertible linear transformations between V and W .
Theorem 1.1.4.Oseledec-Pesin ǫ-reduction theorem ( [Pes76]): Let M be a compact Riemannian manifold.Then there exists a map has the Lyapunov block form: Where D s/u (x) is an s(x)×s(x)/ d−s(x)×d−s(x) block (recall the notation from §0.6 section 4.), respectively.In addition, for every x ∈ N U H χ we can decompose T x M = H s (x) ⊕ H u (x) and R d = R s(x) ⊕ R d−s(x) , and C χ (x) sends each R s(x)/d−s(x) to H s/u (x).Furthermore, ∃κ(χ, f ) s.t.
Proof.(Compare with [KM95] §S.2.10) Define: Notice that for a fixed point x ∈ N U H χ , the sum converges locally-uniform ly on H s (x), H u (x).Define: where u, v ∈ x x M, and π s/u : T x M → H s/u (x) are the orthogonal projection Choose measurably 1 C χ (x) : T x M → R d to be a linear transformation fulfilling u, v ′ x = C −1 χ (x)u, C −1 χ (x)v and C χ [H s (x)] = R s(x) (and hence C χ [H u (x)] = (R s(x) ) ⊥ ).That means D χ is in this block form since C χ [H s ] ⊥ C χ [H u ] by definition of •, • .We get: ), so: where κ = max{e χ (1 The same holds for D u : and since v u , v u ′ x,u > 2|v u | 2 : dxf vu,dxf vu |vu| ′ x,u < dxf vu,dxf vu 2|vu| , we get: Let w s ∈ R s(x) , and define v s := C χ (x)w s : The same way goes to show that |Du(x)wu| 2 |wu| 2 ∈ [e 2χ , κ 2 ].
Claim 1.1.5.D χ (x) extends to a cocycle the following way: D χ and df have the same Lyapunov spectrum.
1 By the orientability of M , there exist a measurable family of positively oriented and orthonormal (w.r.t to the Riemannian metric) bases of TxM , ( ẽ1 (x), ..., ẽd (x)).Using projections to H s/u (x) (which are defined measurably), and the Gram-Schmidt process, we construct orthonormal bases for H s/u (x) in a measurable way, w.r.t to •, • ′ x,s/u .Denote these bases by {b i (x)} s(x) i=1 and {b i (x)} d i=s(x)+1 , respectively.Define C −1 χ (x) : b i (x) → e i , where {e i } d i=1 is the standard basis for R d . Proof.
(∵ the spectra of D χ and df are equal and cancel each other) And by remembering C −1 χ (•) ≥ 1 we get that the limit-inferior of the same expression is greater or equal to zero. By we get the other sided ineq.for C χ (f n (x)).Similar arguments are being done for the liminf and limsup of C χ (x) .
Theorem 1.1.7.There exists an F 0 > 0 only depending on f, χ and M such that: where, | • |-when applied on a tangent vector at x-is the Riemannian norm on T x M , and Claim 1 : For a fixed x ∈ N U H χ the RHS expression is continuous in ξ, η.
Proof : The norms of vectors clearly are.So we are left to show continuity for S 2 , and it's done similarly for U 2 : Denote S 2 (x) = sup v∈H s (x)(1) S 2 (x, v) (recall H s (x)(1) =unit ball in H s (x)).We first show that S 2 (x) is finite: Choose some orthonormal basis (g i ) i for (for an instance) H s (x), and let v ∈ H s (x)(1), then v = s(x) a i g i .Thus, where ) for all m.Substituting these completes the proof that S 2 (x) is finite.Now for continuity, denote v = u + h, then and this concludes claim 1.
A continuous function on a compact set achieves its maximum.So denote by ξ and η the tangent vectors that achieve the maximum at the point f (x): Claim 2 : The left fraction of (3) is bounded away from 0 and ∞ uniformly by a constant (and its inverse) depending only on f, M and χ.Proof : Thus, this, together with noticing that S 2 (f (x), A similar calculation is done for U , and this is enough to write the terms in the numerator with those of the denominator times an independent constant.
Claim 3 : The right fraction of (3) is bounded away from 0 and ∞ uniformly by a constant (and its inverse) depending only on f, M and χ.
Proof : By recalling that The three claims together complete the upper bound for the theorem.By choosing the vectors ξ, η to maximize the denominator instead, we get a lower bound.
Lemma 1.1.8.Let (X, B, µ, T ) be a probability preserving transformation (p.p.t).Let f : X → GL d (R) be a measurable function.Then for almost every x ∈ X there exists a subsequence of the naturals, n k ↑ ∞, s.t.
) is metric and seperable w.r.t the desired topology), this set is of full measure.This is the set we will work with from now on.
So the following Lyapunov limits agree: • lim and the Lyapunov spaces are: (We already knew the Lyapunov exponents are the same) We will start by noticing that: So we get: The last inequality is due to the fact that w s ⊥w u .
1.1.3.Pesin charts.Let exp x : T x M → M be the exponential map.Since M is compact, there exist r, ρ > 0 x (y) is well defined and 2-Lipschitz on B ρ (z) × B ρ (z) for z ∈ M , and so small that This definition is a bit different than Sarig's, and will come in handy in some arithmetics later on.
Remark: Depending on context, the notations of (a) will be used to describe balls with the same respective norms in some subspace of R d -usually R s(x) or its orthogonal complement, for some Theorem 1.1.13.(Compare with [BP07], theorem 5.6.1)For all ǫ small enough, and x ∈ N U H * χ : (1) ψ x (0) = x and ψ x : R Qǫ(x) (0) → M is a diffeomorphism onto its image s.t.d u ψ x ≤ 2 for every u ∈ R Qǫ(x) (0).(2) f x is well defined and injective on R Qǫ(x) (0), and (a) Proof.This proof is similar to Sarig's [Sar13], theorem 2.7; up to a change of a few constants to fit the general dimension d case.
So by Poincaré's recurrence theorem: and for all x ∈ B m : Definition 1.1.17.
By the previous lemma, the above set is of full measure.This is the set we will be interested in from now on.

Proof. Suppose ψ η
x and ψ ζ y ǫ-overlap, and fix some D ∈ D which contains x, y such that d(x, y) Our first constraint on ǫ is that it would be less than 1, and so small that where r(M ), ρ(M ) are defined on §1.1.3,and (1) L 1 is a uniform Lipschitz constant for the maps (x, v) → (exp We assume WLOG that these constants are all larger than one.
The rest of the proof is similar to the proof of [Sar13], proposition 3.2, with only a few changes in the constants to accommodate the general dimension d.
Remark: By 2. the greater the distortion of ψ x or ψ y the closer they are one to another.This distortion compensating bound will be used in the sequel to argue that ψ −1 f (x) •f •ψ x remains close to a linear hyperbolic map if we replace ψ f (x) by an overlapping map ψ y (prop.1.2.4 below).
This lemma is conceptually similar to lemma 1.2.3 in [Sar13], with a similar proof.We make a few changes in the formulation to fit our own technical needs, as we discard the use of the terms "distortion bounds" as it is being done in Sarig's work.
Proof.Following the notations of the last proposition, we denote C x/y with C 1/2 ("/" is used here as in the notations of §0.6, section 4): Its derivative at the origin is : ).This concludes part 1.
For part 2: And by symmetry 1.2.2.The form of f in overlapping charts.We introduce a notation: for two vectors u ∈ R d1 and v ∈ R d2 , (−u−, −v−) ∈ R d1+d2 means the new vector whose coordinates are the coordinates of these two, put in the same order as written.
The second and fourth points imply that according to proposition 1.2.2.Therefore f xy is well-defined, differentiable and injective on R Qǫ(x) (0).h s is defined by taking the first S coordinates of f xy (v s , v u ), and subtracting D s v s : basically the equation in the statement defines h s , h u .It is left to check the properties: x The norm of the first summand is bounded by 3κǫη.The norm of the second summand is less than ǫ|v| β/2 < √ dǫη β/2 .The third term is , where, then ∂(hs,hu) ∂(vs,vu) < ǫη β/3 on R η (0).In particular: ∂(hs,hu) ∂(vs,vu) | 0 < ǫη β/3 .From the expression for d v f xy we get that for every v 1 , v 2 ∈ R Qǫ(x) (0): 0), and plugging in: 1.2.3.Coarse graining.Recall the definition of s(x) for a Lyapunov regular x ∈ N U Hχ, in definition 1.1.1,and the assumption s(x) ∈ {1, ..., d − 1}.
Proposition 1.2.5.The following holds for all ǫ small enough: There exists a countable collection A of pesin charts with the following properties: (1) Discreteness: {ψ η x ∈ A : η > t} is finite for every t > 0 (2) Sufficiency: For every x ∈ N U H * χ and for every sequence of positive numbers 0 < η n ≤ e −ǫ/3 Q ǫ (f n (x)) in I ǫ s.t. e −ǫ ≤ ηn ηn+1 ≤ e ǫ there exists a sequence {ψ ηn xn } nZ of elements of A s.t. for every n: (a) ψ ηn xn ǫ-overlaps ψ ηn f n (x) , Qǫ(f n (x)) Qǫ(xn) = e ±ǫ and s(x n ) = s(f n (x)) (= s(x), since Lyapunov exponents and dimensions are f -inv.)(b) ), e ǫ η n } Proof.See [Sar13], prop.3.5.Adjust to the fact that in the general dimensional case, the dimension of the stable Lyapunov subspace can vary from point to point.We address this in the discretization we do, and in the properties our definitions, and thus our constructions, posses.We define , and a real number η.A chain of charts also posses the same stable dimension s(x) for each term.The proper discretization of the space X is thus the key for the coarse graining and this proposition.
(this requirement did not appear in Sarig's definition, as it is trivial in the two dimensional hyperbolic case) for all i.We abuse terminology and drop the ǫ in "ǫ-chains".
Let A denote the countable set of Pesin charts we have constructed in §1.2.3 and recall that This is a countable directed graph.Every vertex has a finite degree, because of the following lemma and prop.1.2.5 .
is the Markov-shift associated with the directed graph G.It comes equipped with the left-shift σ, and the standard metric.
Lemma 1.3.4.If ψ p s ,p u x → ψ q s ,q u y then q u ∧q s p u ∧p s = e ±ǫ .Therefore for every ψ p s ,p u x ∈ V there are only finitely many ψ q s ,q u y ∈ V s.t.ψ p s ,p u x → ψ q s ,q u y or ψ q s ,q u y → ψ p s ,p u x .
) is equal to Q n for infinitely many n > 0, and for infinitely many n < 0.
for every k ∈ Z. Proof.See proposition 4.5 in [Sar13], and adjust for the possibility of different dimensions of stable Lyapunov subspaces, for different regular points (which may occur in the general dimensional case).The adjustment can be done by noticing that for our case, as in Sarig's case and proof, the overlap condition is monotone.

Admissible manifolds and The Graph Transform.
Definition 1.3.9.
where 0 < q ≤ Q ǫ (x), and Similarly we define an s− manifold in ψ x : with the same requirements for − → F s and q.We will use the superscript "u/s" in statements which apply to both the s case and the u case.The function The parameters of a u/s manifold in ψ x are (in this context for a matrix A: where Remark: Notice that the dimensions of an s or a u manifold in ψ x depend on x, and that they sum up to d.
, and q = p u u − manifolds p s s − manifolds.
Remark: If ǫ < 1 (as we always assume), then these conditions together with p u , p < ǫ and by Lagrange's mean-value theorem applied to the restriction of F to the interval connecting each t 1 and t 2 , we are done.Another important remark: If ǫ is small enough then max Definition 1.3.11. ( where Dom is the closure of some open and bounded subset of , where, recall, Remark: (1) The second section of this definition is a generalization of the specific β 3 -norm of admissible manifolds, as represented by the σ parameter in definition 1.3.9.
(2) Notice that the difference between • and • ∞ in our notations, is that • is taken over an operator, while • ∞ is taken over a map that returns operators, and returns the supremum of norms over its whole domain.
Lemma 1.3.12.Let E : dom(E) → M r (R) be a map as in definition 1.3.11.Then for any 0 Proof.We claim that for any two ϕ and ψ satisfying the same assumptions as It is also easy to check that BH := {ϕ : dom(E) → M r (R) : ϕ α < ∞} is a complete Banach space, and hence a Banach algebra.By a well known lemma in Banach algebras, ( and we are done.
Proposition 1.3.13.The following holds for all ǫ small enough.Let V u be a u−admissible manifold in ψ p u ,p s x , and V s an s−admissible manifold in ψ p u ,p s x , then: (1) V u and V s intersect at a unique point P 2 ).Remark: We will omit the − → • notation when it is clear that the object under inspection is a vector.Write Let η := p s ∧ p u .Note that η < ǫ and that |F (0)| ∞ , |G(0)| ∞ ≤ 10 −3 η and Lip(F ), Lip(G) < ǫ by a remark following definition 1.3.10.Hence the maps F and G are contractions, and they map the closed cube R 10 −2 η (0) (in the respective dimensions) into itself, because for every (H, t) ∈ {(F, t u ), (G, t s )}: We claim that P is the unique intersection point of V s , V u .Denote ξ = p u ∨ p s and extend F, G arbitrarily to ǫ−Lipschitz functions F , G : R ξ (0) → R Qǫ(x) (0) [HU80].This can be done using McShanen's extension formula f (x) = sup Any intersection point of V s , V u is an intersection point of F , G, which takes the form P = ψ x (−ṽ−, − w−) where ṽ = F ( w) and w = G(ṽ).Notice that w is a fixed point of G • F .The same calculations as before show that G • F contracts R ξ (0) into itself.Such a map has a unique fixed pointhence w = w, whence P = P .This concludes parts 1,2.Next we will show that P is a Lipschitz function of (V s , V u ).Suppose V u i , V s i (i = 1, 2) are represented by F i and G i respectively.Let P i denote the intersection points of V s i ∩ V u i .We saw above that and passing to the limit n → ∞: and in the same way max 1−ǫ 2 < 3, and this completes part 3.
The next theorem analyzes the action of f on admissible manifolds.Such analysis is often referred as a collection of Graph Transform Lemmas, and is used to show Pesin's stable manifold theorem ([BP07] chapter 7, [Pes76]).The earliest version of these methods were introduced firstly by Perron in [Per29,Per30].The analysis we make shows that the Graph Transform preserves admissibility as defined above.The general idea is similar to what Sarig does in [Sar13], §4.2 for the two dimensional case.Katok and Mendoza treat the general dimensional case in [KM95], but our case requires more strict additional analytic properties, as in Sarig's work.Our work also requires additional analysis to accommodate the vector calculus case, and some variations to claims.
Theorem 1.3.14.(Graph Transform) The following holds for all ǫ small enough: suppose ψ p s ,p u x → ψ q s ,q u y , and V u is a u-admissible manifold in ψ p s ,p u x , then: intersects any s-admissible manifold in ψ q s ,q u y at a unique point.
(3) V u restricts to a u-manifold in ψ q s ,q u y .This is the unique u-admissible manifold in Similar statements hold for the f −1 -image of an s-admissible manifold in ψ q s ,q u y .
Proof. .We omit the u super script of F to ease notations, and denote the parameters of V u by σ, γ, ϕ and q, and let η := p s ∧ p u .V u is admissible, so σ ≤ 1 2 , γ ≤ 1 2 η β/3 , ϕ ≤ 10 −3 η, q = p u , and Lip(F 0).On this domain, f xy can be written as follows (by prop.1.2.4): . From the proof of prop.1.2.4 we got that and |h s/u (0)| < ǫ 2 on graph(F ).Using the equation for f xy we can put Γ u y in the following form: The idea (as in [KM95])im is to call the "u" part of coordinates τ , solve for t = t(τ ), and to substitute the result in the "s" coordinates.
Claim 1 : The following holds for all ǫ small enough: D u t + h u (F (t), t) = τ has a unique solution t = t(τ ) for all τ ∈ R e χ− √ ǫ q (0), and For every |t| ∞ ≤ q and a unit vector v: Since v was arbitrary, it follows that τ is expanding by a factor of at least e −ǫ D −1 We estimate this set: Since τ is continuous and e −ǫ D −1 u −1 expanding (recall, these calculations are done w.r.t the supremum norm, as defined in definition 1.3.9): ).The center of the box can be estimated as follows: Recall that η ≤ q, and therefore |τ (0)| ∞ ≤ 2ǫq, and hence ⊃ R e χ e − √ ǫ q (0) for ǫ small enough.So τ −1 is welldefined on this domain.
We saw above that |τ (0)| ∞ < 2ǫη.For all ǫ small enough, this is significantly smaller than e χ− √ ǫ q, therefore τ (0) belongs to the domain of t.It follows that: For all ǫ small enough this is less than 2ǫη, proving (b).
Next we will calculate the β 3 -norm of d s t.We require the following notation: G(s) := (F (t(s)), t(s)), and notice that d G(s) h u is a u(x) × d rectangular matrix, where u(x) := d − s(x).
, and A(s where the notation in the definition of A ′ means that it is the matrix created by stacking the two matrices d t(s) F (represented in the standard bases, as implied by the notation) and I u(x)×u(x) .Using this and the identity we get: From lemma 1.3.12we see that in order to bound the β 3 -norm of d • t, it's enough to show that the β 3 -norm of D −1 u A(•) is less than 1-the lemma is applicable since d s t is, indeed a square matrix as required, as both t and s in this set up are vectors in ∞ Now we shall use the following previously shown facts: From the identity G(s) = (F (t(s)), t(s)) we get that Lip(G)≤ Lip(F ) • Lip(t) + Lip(t); and from that and the first two items above we deduce that Lip(G) ≤ 1.We also know From the last two items we deduce d . From the admissibility of the manifold represented by F , we get Höl β 3 (F ).So it is clear that for a small enough ǫ, Höl β/3 (A(•)) < 2ǫ 2 , and hence Höl , and as quoted before claim 1 from prop.1.2.4 d G(s) h u ≤ ǫ 2 .So together with the conclusion d • F ≤ ǫ we can deduce that for small enough ǫ, A ′ (0) ≤ ǫ 2 (1 + ǫ) < 2ǫ 2 , and Thus we can use lemma 1.3.12.Using this lemma, together with the estimation we have just got, on Eq. ( 5), yields for a small enough ǫ, since D u is at least e χ -expanding.This completes claim 1.
We now return to substituting t = t(s), we find that where s ∈ R u(x) (as it is the right set of coordinates of (H(s), s)), and H(s) := D s F (t(s)) + h s (F (t(s)), t(s)).
Claim 4 : For all ǫ small enough d Proof : By claim 1 and its proof: We shall use the small fact shown in the proof of lemma 1.3.12:For any two ψ, ϕ : Dom → M r1 (R), where Dom is the closure of some open and bonded subset of R r2 and r 1 , Using the small fact from above for the left summand, and our bound for A(•) β 3 we derived in claim 1 for the right summand, we get d • H β/3 ≤ D s • σ • e 3ǫ−χ + 4ǫ 2 .This and D s ≤ e −χ give us: The parameters of V u satisfy section 1 from the statement of the theorem, and V u contains a u-admissible manifold in ψ q u ,q s y .Proof : To see that V u is a u-manifold in ψ y we have to check that H is Consequently, for every ǫ small enough So V u restricts to a u-manifold with q-parameter equal to q u .Claims 2-4 guarantee that this manifold is u-admissible in ψ q s ,q u y , and that part 1 of theorem 1.3.14holds.
Proof : The previous claim shows existence.We prove uniqueness: using the identity We just saw in the end of claim 5 that for all ǫ small enough, q u < e χ− √ ǫ q.By claim 1 the equation τ = D u t + h u (F (t), t) has a unique solution t = t(τ ) ∈ R q (0) for all τ ∈ R q u (0).Our manifold must therefore equal ] This is exactly the u-admissible manifold that we have constructed above. Let By the previous paragraph it is enough to show that the second set of coordinates (the u-part) of ψ −1 y [{f (p)}] has infinity norm of less than q u .Call the u-part τ , then: This concludes the claim.
Claim 7 : f [V u ] intersects every s-admissible manifold in ψ q s ,q u y at a unique point.Proof : In this part we use thoroughly the fact that in our definition for ψ p s ,p u x → ψ q s ,q u y s(x) = s(y).Let W s be an s-admissible manifold in ψ q s ,q u y .We saw in the previous claim that f [V u ] and W s intersect at least in one point.Now we wish to show that the intersection point is unique: Recall that we can put we saw in the proof of claim 1 that the second coordinate τ (t) := D u t + h u (F (t), t) is a 1-1 continuous map whose image contains R q u (0).We also saw that d Consequently the inverse function t : Im(τ ) → R q (0) satisfies d • t < 1, and so where Lip(H) ≤ ǫ Let I : R q u (0) → R u(x) denote the function which represents W s in ψ y , then Lip(I) ≤ ǫ.Extend it to an ǫ-Lipschitz function on Im(τ ) (again, by [HU80]).The extension represents a Lipschitz manifold W s ⊃ W s .We wish to use the same arguments as we used in proposition 1.3.13 for uniqueness of intersection point (this time of f [V u ] and W s )-it requires the following observations: (1) R q u (0) ⊂ R e χ− √ ǫ q (0) ⊂ Im(τ ), as seen in the estimation of Im(τ ) in the beginning of claim 1.
(2) Im(τ ) is compact, as a continuous image of a compact set. (3) the last inclusion is due to the bound we have seen |τ (0)| ≤ 2ǫη) Hence, as in the arguments of proposition 1.3.13,we can use the Banach fixed point theorem the same way (for a contracting continuous function from a compact set into itself).This gives us that in particular, f [V u ] and W s intersect in one point at most.This concludes this claim, and thus the proof of the theorem.The case of the s-manifold follows from the symmetry between s and u-manifolds: (1) V is a u-admissible manifold w.r.t f iff V is an s-admissible manifold w.r.t to f −1 , and the parameters are the same.(2) ψ p s ,p u x → ψ q s ,q u y w.r.t f iff ψ q s ,q u y → ψ p s ,p u The operators F s/u depend on the edge ψ ps,pu x → ψ qs,qu y .
Proposition 1.3.16.If ǫ is small enough then the following holds: For any s/u-admissible manifold Proof.(For the u case) Suppose ψ p s ,p u x → ψ q s ,q u y , and let V u i be two u-admissible manifolds in ψ p s ,p u x .We take ǫ to be small enough for the arguments in the proof of theorem 1.3.14 to work.These arguments give us that if where t i and τ are of length u(x) (as a right set of components) and In order to prove the proposition we need to estimate The claim follows.
Proof : 1) Now when ǫ → 0 the RHS multiplier goes to 0, hence for small enough ǫ it is smaller than e −χ/2 .Part 3 : For all ǫ small enough: We have seen at the beginning of the Graph Transform that Id = (D u + A i (s))d s t i .By taking differences we obtain: , which is the same as A ′ i (s), with a parameter change (t = t i (s)).So now we see that: 3ǫ 2 , and hence: ∞ , it follows that: < 1 (from the definition of admissible manifolds and the proof of the Graph Transform) we get that: Taking differences we that: Using the same arguments as in part 3 we can show that: ∞ (same as in the estimate of I in part 3) Recalling the definitions from the begining of §1.3: is the u-admissible manifold in v 0 which is a result of the application of the graph transform n times along v −n , ..., v −1 (each application is the transform described in section 3 of theorem 1.3.14).Similarly any s-admissible manifold in v n is mapped by n applications of the graph transform to an s-admissible manifold in v 0 : F n s (v n ).These two manifolds depend on v −n , ..., v n .Definition 1.3.19.Let ψ x be some chart.Assume V n is a sequence of s/u manifolds in ψ x .We say that V n converges uniformly to V , an s/u manifold in ψ x , if the representing functions of V n converge uniformly to the representing function of V .
) i∈Z is a chain of double charts, and choose arbitrary u-admissible manifolds V u −n in v −n , and arbitrary s-admissible manifolds V s n in v n .Then: (1) The limits Proof.Parts (1)-(4) are a modification of Pesin's Stable Manifold Theorem [Pes76], and Sarig's version of it.Part (5) is a modification of Brin's Theorem on the Hölder continuity of the Oseledets distribution [BP01].Whereas Brin's theorem only states Hölder continuity on Pesin sets, part (5) gives Hölder continuity everywhere, as in Sarig's version.
Parts (1), (3), (4) and ( 5) are quite similar to the proof of proposition 4.15 in [Sar13], we nonetheless present the proof since the understanding of this proposition is important to the ideas presented in this work.
We show a proof for the case of u manifolds, the stable case is similar:

By the previous proposition for any other choice of u-admissible manifolds
(Since for any admissible manifold the Hence, this is a Cauchy sequence in a complete space, and therefore converges.
Part 2 : Admissibility of the limit: Write v 0 = ψ p s ,p u x and let F n denote the functions which represent For each term in the sequences, the following identity holds: In fact the same calculations give us that ∀s ∈ R q u (0) : Since L is β/3-Hölder on a compact set, and in particular uniformly continuous, the second summand is a o(|t − s|), and hence ∃d t F = L(t).We also see that {d • F n } can only have one limit point.Consequently ).The first and third summands tend to zero by definition, and the second goes to zero since Hence by part 3 for every k ≥ 0: ] for all k ≥ 0. Write z = ψ x0 (v 0 , w 0 ).We show that z ∈ V u by proving that v 0 = F (w 0 ) where F is the representing function for V u : For this purpose we will use z ] (the first point by assumption, and the second since It is therefore possible to write: , and max we get the two following bounds: and assume that ǫ is small enough that e −χ/2 + ǫ ≤ e −χ/3 , e χ/2 − ǫ ≥ e χ/3 then we get (since b 0 ≡ 0): Proof by induction: it is true for k = 0 since b 0 = 0. Assume the induction step, then: We see that a k+1 ≥ (e χ/3 − ǫ)a k for all k ≥ 0, hence a k ≥ (e χ/3 − ǫ) k a 0 .Either a k → ∞ or a 0 = 0.But , so a 0 = 0, and therefore Which concludes this part.
Let F N , G N be their representing functions.Admissibility implies that F k and G k : by the previous proposition we get ∞ ) Iterating the first inequality, from k = N , going down, we get: Passing to the limit as n → ∞ we get: hence (by induction): ).Now substitute k = N , and remember ) and V u ((w i ) i≤0 ) (resp.) in C 1 .Therefore if we pass to the limit as n → ∞ we get and π[Σ # ] have full probability w.r.t any hyperbolic invariant measure with Lyapunov exponents bounded from 0 by χ.
Proof.Part 1 : π is well-defined thanks to proposition 1.3.13.Now, write v i = ψ ), and that is an . For k > 0 we use the previous proposition (part 3) to see that The case k < 0 can be handled the same way, using V u ((v i ) i≤0 ).Thus z satisfies: Any point which satisfies this must be z, since by the previous proposition (part 4) it must lay in V u ((v i ) i≤0 )∩ V s ((v i ) i≥0 ), so this equation characterizes π(v) = z.Hence: ∀k and this is the condition that characterizes π(σv).
Part 2 : We saw that v → V u ((v i ) i≤0 ) and v → V s ((v i ) i≥0 ) are Hölder continuous (previous proposition).Since the intersection of an s-admissible manifold and a u-admissible manifold is a Lipschitz function of those manifolds (proposition 1.3.20),π is also Hölder continuous.
, then by the last proposition there's a ψ for all k, and s.t.ψ ) i∈Z .In fact prop.1.3.8gives a chain s.t.
. By the definition of N U H # χ (f ) there exists a sequence i k , j k ↑ ∞ for which p s i k ∧ p u i k and p s −j k ∧ p u −j k are bounded away from zero.By the discreteness property of A (prop.1.2.5), ψ must repeat some symbol infinitely often in the past, (possibly a different symbol) in the future.Thus the above actually proves that Define this as all 'regular' chains of Σ.
Proof.All the properties of π : Σ rel → M are obvious, besides the statement that π 2. Regular chains which shadow the same orbit are close 2.1.The inverse problem for regular chains.The aim of this part is to show that the map π : Σ # rel → N U H χ from theorem 1.3.21, is "almost invertible".Meaning, we aim to show that if π((ψ ) i∈Z ), then those two chains must be close-∀i: -the respective parameters belong to the same compact sets.The compact sets can be as small as we wish by choosing small enough ǫ.The discretization we constructed in §1.2.3 gives that in fact the compact sets contain a finite number of such possible parameters for our charts.Moreover, we will get that π can be refined to be finite-to-1.
Definition 2.1.2.Let V u be a u-admissible manifold in the double chart ψ p s ,p u x .We say that V u stays in windows if there exists a negative chain (ψ Definition 2.1.3.Let V s be an s-admissible manifold in the double chart ψ p s ,p u x .We say that V s stays in windows if there exists a positive chain (ψ x and s-admissible manifolds W s i in ψ Proposition 2.1.4.The following holds for all ǫ small enough: Let V s be an admissible s-manifold in ψ p s ,p u x , and suppose V s stays in windows. (1) For every y, z Notice that from the third list item, we get in particular that ∀u ∈ T y V s : S(y,u) S(z,Λu) , |u| |Λu| = e ± 1 3 Qǫ(x) β/4 .The symmetric statement holds for u-admissible manifolds which stay in windows, for "s" replaced by "u" and f by f −1 .
Proof.Suppose V s is an s-admissible manifold in ψ p s ,p u x0 , which stays in windows, then there is a positive chain (ψ , and there are s-admissible manifolds W s i in ψ for all i ≥ 0. We write: s/u ≤ 3ǫ2 .Thus: , their stable components are in R p s 0 (0), so |y 0 − z 0 | ≤ 2p s 0 .So we get: Part 2 : We assume ǫ is small enough that e −χ + 3ǫ 2 ≤ e − 2χ 3 .Let y ∈ V s and let u ∈ T y V s (1) some unit where the left multipliers on the RHS are block matrices as denoted.So we get: Returning to the defining relation , and recalling that d • ψ x k ≤ 2 we see that: Part 3 : There exists a unique In this part we reassign the symbols ξ k , η k .First, define: is the projection into the first s(x) coordinates; and is identified with its differential.We claim that: for some constant C(χ, β) depending only on χ and β Proof: Define We call the left term on the RHS M k+1 , and estimate it: Remark: the norm of the first summand can actually be taken just over tangent vectors of the form ξ d y k F k ξ ; and in lemma 1.2.3 we actually saw d , so we can continue to bound the expression the following way: So for small enough ǫ this is less than e −2χ/3 .Thus: Notice that by the bound we have found for 3 k (and the same for z ′ 0 ).Substituting this into the bound we found for M k+1 gives and by recursion The penultimate transition is due to the equality max x≥0 {e −αx x} = 1 eα , and the ultimate transition is due to the inequality 6 e e χ 6 ≤ 3, for small enough χ.
We estimate By the definition of ψ x k , and since So the denominators are bounded below by ) −1 respectively (recall the notations defined in §0.6, section (4)).
Since for every two non-zero vectors u, v: On D k we can write , then: We study this expression.In what follows we identify the differential of a linear map with itself.By construction, the map (x, v, u) Therefore there exists a constant E 0 > 1 s.t. for every (x, v i , u i ) ∈ D × B 2 (0) × B 2 (0) and every D ∈ D: k ) (∵ part 1, and the claim from the beginning of this part) Since (ψ ) i∈Z : p s i = min{e ǫ p s i+1 , Q ǫ (x i )} ≤ e ǫ p s i+1 , whence p s 0 ≤ e kǫ p s k : Plugging this into Eq.( 8) gives: The factor of Q ǫ (x 0 ) β/4 is less than 1 3 for ǫ small enough.How small ǫ should be, depends only on M (through E 0 ), f (through β and H 0 ) and χ.
Definition 2.1.5.Let ψ p s ,p u x → ψ q s ,q u y .Let V s be an s-admissible manifold in ψ q s ,q u y which stays in windows with a representing function F .Let G be the representing function of F (V s ).Notice that by the assumptions for V s and by the graph transform it follows that F (V s ) stays in windows as well.Then (1) p := ψ x (0, G(0)), (2) q := ψ y (0, F (0)).We call these p-points.
Notice: (1) f −1 (q) ∈ F (V s ) (2) These are very lean notations and in fact p = p(V s ), q = q(V s )-It will be clear from context what admissible manifolds corresponds to p or q (3) One can make analogous definitions for the unstable case.
Recall the definition of S(•, •) from the beginning of theorem 1.1.7.Lemma 2.1.6.Let ψ p s ,p u x → ψ q s ,q u y .Let V s be an s-admissible manifold in ψ q s ,q u y which stays in windows.Then ∃π x : where λ = d f −1 (q) f Λξ, ξ ∈ T p F (V s ) arbitrarily and Λ : ) is given by prop.2.1.4.A similar statement holds for the unstable case.We will also see that |πxξ| |ξ| = e ±2Qǫ(x) β/4 for all ξ ∈ T p F (V s ) (similarly for π y ).

2
(x) there exists some D ∈ D containing all of them; Θ would be its associated isometry.
We are left to study the second multiplied term of the RHS of Eq. (10).
This concludes the proof.
Lemma 2.1.7.The following holds for any ǫ small enough.For any two regular chains (ψ for any ξ ∈ T p k V s k , where in the notations of the previous lemma: p k , r k are "p-points" respective to x k , y k , and π x/y,k are π x respective to x k , y k .As in the notations of proposition 2.1.4(3): , which makes the composition Λ y,k Λ x,k well defined.It can be seen by the fact that p is in the image of π, and hence Lyapunov regular; thus f k (p) is Lyapunov regular as well.So H s (f k (p)) must coincide with T f k (p) V s k by proposition 2.1.4(2),since any unstable component to a tangent vector would make the vector unstable.The unstable subspaces must coincide for the same argument in reversed time flow.Thus Proof.(Compare with proposition 7.3 in [Sar13]) Denote v = (ψ ) i∈Z and u = (ψ are the s-admissible manifolds which stay in windows that fulfil the assumptions of lemma 2.1.6.We claim it's enough to prove that: Proving it would be sufficient, since the manifolds V s k stay in windows and contain f k (p), therefore by the third section of proposition 2.1.4: . The same argument applies to U s k .So we can make the following decomposition: So it's clear why that would be enough.To bound that expression: We are assuming v is regular, therefore there exists a relevant double chart v and a sequence Proof : By assumption v is relevant.Choose a chain w s.t.w 0 = v and z := π( w) ∈ N U H χ .W s := V s ((w i ) i≥0 ), denote the "p point" (in the notations lemma 2.1.6) of this manifold with p w .Denote Λ pw,z : T pw W s → T z W s , as in proposition 2.1.4.We know max η S(pw,η) S(z,Λp w ,z η) ≤ e √ ǫ by proposition 2.1.4-itis indeed a maximum and not merely a supremum since S(p w , •) and S(z, Λ pw,z (•)) are continuous as shown in theorem 1.1.7(and Λ pw,z is continuous as a linear map), and it is enough to take the supremum over the compact set η ∈ H s (p w )(1), since we can cancel the size of η from the numerator and denominator of S(pw,η) S(z,Λp w ,z η) .max |η|=1 S(z, Λ pw,z η) < ∞ since z ∈ N U H χ (f ), hence also max |η|=1 S(p w , η) < ∞.So the following quantity is finite and well defined: W s is an admissible manifold in v n k .So by taking W s in V n k+l and applying to it F s n k+l − n k times then the resulting manifolds W s l := F As shown in lemma 2.1.6,the ratio of said S parameters can only improve, so: min η / max η S(t l ,η) S(x,π ] (where t l is the "p" point of W s l , and π (l) x is the respective "π x " linear transformation).The convergence of W s l to V s n k means that if W s l is represented in v n k by F l , and V s n k is represented by F , then F l − F ∞ → 0. Also, since d • F l β/3 are uniformly bounded away from infinity: in the metric of T M ) , and similarly: π (l)  x ξ l → π x ξ (in regular metric of T x M ) In order to be able to conclude from ψ p s ,p u x → ψ q s ,q u y that C −1 χ (x) is similar to C −1 χ (y) (roughly speaking; similarity has not yet been defined rigorously), we need to show that it means that C −1 χ (x), C −1 χ (y) act similarly on the respective stable subspaces.
In the two dimensional case, mapping isometrically (in our case-arbitrarily close to an isometry) the stable subspace of some hyperbolic point to the stable subspace of some other hyperbolic point, is merely choosing a sign from {±1}, since the subspaces are one-dimensional.Hence comparing the actions of different Lyapunov change of coordinates is straightforward.In the multidimensional case, it takes more than choosing a signyou can choose any orthogonal map of the space to itself.Orthogonal maps do not necessarily commute.Thus, choosing the linear mapping can affect the way the Lyapunov change of coordinates acts.
We construct a mapping between the stable subspaces which keeps the action of the Lyapunov change of coordinates similar, and allows the comparison.
Claim 2.1.8.Under the assumptions of the previous lemma: where Ξ i is an invertible linear transformation, and Ξ i ≤ exp(ǫ).
Proof.Define Ξ k the following way on the stable/unstable spaces: where π x/y,k and Λ x/y,k are defined as in lemma 2.1.7.The composition of Λ y,k Λ x,k is well defined by the remark after lemma 2.1.7.The properties of the composing elements of Ξ k imply that it preserves the stable and unstable subspaces.Ξ k extends linearly to the whole tangent space of x k from the definition on the stable and unstable subspaces, using the requirement of being linear.Hence, from definition, and lemma 2.1.7: where ξ = ξ s + ξ u , ξ s/u ∈ H s/u (x k ).We begin by showing bounds on norms of Ξ k , Ξ −1 k when restricted to the stable/unstable subspaces.By the bounds for the restricted π x , π −1 x ; Λ , Λ −1 from lemma 2.1.6and prop.2.1.4resp., )) (14) Now to bound these norms on the whole tangent space: To ease notations we will omit the 'k' subscripts.In what is to come, / is as in the notations of §0.6, section 4: d(x/y, f k (p)) < Q ǫ (x/y) < ǫ < ρ (by the definition of Q ǫ (•) in def.1.1.12).Hence, the following is well defined: ψ −1 x (f k (p)) =: z ′ x , ψ −1 y (f k (p)) =: z ′ y .The vectors of the first s coordinates of z ′ x/y will be called z x/y respectively (the same notations as before).
Part 1: Let ξ ∈ T x M , ξ = ξ s + ξ u , ξ s/u ∈ H s/u (x)-then |ξ s/u | ≤ C −1 χ (x) • |ξ|.Proof : WLOG |ξ| = 1.We wish to give a bound to the size of ξ s/u that can be very big, even when |ξ s + ξ u | = |ξ| = 1.This can happen when the angle between the stable and unstable spaces ∢(H s (x), H u (x)) = inf ηs∈H s (x),ηu∈H u (x) |∢(η s , η u )| is very small, and ξ s and ξ u are almost parallel, of the same size, and are pointing to almost opposite directions.Thus their sum cancels each other almost completely and leaves a small remainder vector.They can be very big, while their sum is of size 1.
This shows the wished inequality.Thus in total, |ξ s |, |ξ u | ≤ C −1 χ (x) .Part 2: Ξ = d Cχ(x)z ′ x (exp −1 y exp x ) + E 1 , where E 1 is linear and E 1 ≤ Q ǫ (x) Proof : We will begin by showing that the expression above is well defined for ǫ < ρ 2 .This is because exp x [B ǫ (0)] ⊂ B ǫ+d(x,y) (y) ⊂ B 2ǫ (y); |C χ (x)z ′ x | = d(x, f k (p)) < ǫ; so exp −1 y exp x is defined on an open neighborhood of C χ (x)z ′ x , and hence is differentiable on it.By the definition of Ξ, on each stable/unstable subspace it acts as a composition of two linear maps: • At first ξ ∈ H s (x) (WLOG) is mapped to ξ ′ a tangent vector at f k (p).By the definition of the components of Ξ: , where G is representing the appropriate stable manifold, and v = C −1 χ (x)ξ.ξ ′ has the following expansion (we write d z ′ y ψ −1 y , without specifying the meaning is (d z ′ y ψ y ) −1 ): The last equality is due to the fact that we know ξ ′ is tangent to the stable manifold in ψ Now using this again in Eq. ( 13): . By symmetry we will also get C −1 χ (y) ≤ 5 C −1 χ (x) .Substituting these in the bounds for E 1 : Since V s and U s intersect, also f n [V s ] and f n [U s ] intersect.We use the assumption that p s ≤ q s to show that S v ⊂ S u .Otherwise there is a point z ∈ ∂S u s.t.z is in the relative interior of S v .This can be seen by constructing a continuous path included in the interior of S v connecting some z ′ ∈ S u ∩ S v with some point in S v \ S u ( = φ by assumption); there is such a path since V s is a connected manifold and S v is a homeomorph of V s .z is the intersection of the path γ and ∂S u .I.e.denote the continuous path by γ : [0, 1] → S v , γ(0) = z ′ , then t 0 := inf{t ∈ [0, 1]|γ(t) ∈ S u } ⇒ z := γ(t 0 ).
Then since f is a homeomorphism: f −n (ψ xn (z, G(z))) is in the relative interior of V s .Now, since U s is a closed set, so is S u ; and the fact that t → f −n (ψ xn (t, G(t))) : S u → U s is continuous and onto U s gives us that z being a boundary point of S u means that f −n (ψ xn (z, G(z))) must be a boundary point of U s as well.So we get that U s has a boundary point in the relative interior of V s .We now use the assumption that x = y and view V s and U s as submanifolds of the chartψ p s ,p u x .The boundary points of U s have s-coordinates with | • | ∞ -norm equal to q s , and the points of V s have s-coordinates with | • | ∞ -norm less or equal to p s .It follows that q s < p s , in contradiction to our assumption.It follows that f n [V s ] ⊂ f n [U s ], whence V s ⊂ U s .

0. 3 .
Comparison to other results in the literature.
y) < ρ}, where d(•, •) is the distance function on M × M , w.r.t to the Riemannian metric.Definition 1.1.12.(a)B η (0), R η (0) are open balls located at the origin with radius η in R d w.r.t Euclidean norm and supremum norm respectively.(b) We take ρ so small that (x, y) → exp −1

1.2. 1 .
The overlap condition.The following defintion is from [Sar13], §3.1.Definition 1.2.1.(a) For every x ∈ M there's an open neighborhood D of diameter less than ρ and a smooth map Θ D : T D → R d s.t.: be a finite cover of M by such neighborhoods.Denote with λ(D) the Lebesgue number of that cover.Hence if d(x, y) < λ(D) then x and y belong to the same D for some D. (b) We say two Pesin charts ψ η1 x w.r.t f −1 Definition 1.3.15.Suppose ψ ps,pu x → ψ qs,qu y .The Graph Transforms are the maps F s/u which map a s/u-admissible manifolds V s/u in ψ qs,qu y /ψ ps,pu x to the unique s/u-manifold in ψ ps,pu x /ψ qs,qu y equipped with the left-shift σ, and the metric d( v, w) = e − min{|i|:vi =wi} Definition 1.3.18.Suppose ( 2 we have covered M by a finite collection D of open sets D, equipped with a smooth map θD : T D → R d s.t.θ D | TxM : T x M → R d is an isometry and ν x := θ −1 D : R d → T M has the property that (x, v) → ν x (v) is Lipschitz on D × B 1 (0).Since f is C 1+β − and M is compact, d p f vdepends in a β-Hölder way on p, and in a Lipschitz way on v.It follows that there exists a constant H 0 > 1 s.t. for every D ∈ D; y, z ∈ D; u, v ∈ R d (1): log |d y f ν y