Tangent bundles to regular basic sets in hyperbolic dynamics

Given a locally maximal compact invariant hyperbolic set $\Lambda$ for a $C^1$ flow or diffeomorphism on a Riemann manifold with $C^1$ unstable laminations, we construct an invariant continuous bundle of tangent vectors to local unstable manifolds that locally approximates $\Lambda$ in a certain way.


Introduction
Let M be a C 1 complete (not necessarily compact) Riemann manifold M and let Λ be a basic set for a C 1 flow φ t : M −→ M or a C 1 diffeomorphism f : U −→ f (U ) ⊂ M , where U is an open neighbourhood of Λ in M . Let E u (x) and E s (x) (x ∈ Λ) be the tangent spaces to the local stable and unstable manifolds W s ǫ (x) and W u ǫ (x) of size ǫ > 0, respectively (see Sect. 2). We say that φ t has C 1 unstable laminations over Λ if the map Λ ∋ x → E u (x) is C 1 , i.e. at each x ∈ Λ this map has a linearization that depends continuously on x.
Given x ∈ Λ, let k x ≥ 1 be the minimal integer such that there exist δ x ∈ (0, ǫ) and a k xdimensional C 1 submanifold S x of W u ǫ (x) such that Λ ∩ W u δx (x) ⊂ S x . We will then say that S x is a u-submanifold of minimal dimension at x and size δ x . In general S x is not (locally) unique, however it turns out that, under certain regularity conditions, its tangent space T x S x is uniquely determined, invariant and continuous: Theorem 1. Assume that Λ is a basic set for the flow φ t (or the diffeomorphism f ) and the local unstable laminations over Λ are C 1 . Then there exists an integer k = k u ≥ 1 and a continuous dφ t -invariant (or df -invariant, respectively) distribution E u Λ (x), x ∈ Λ, of k-dimensional linear spaces such that for any x ∈ Λ we have k x = k, E u Λ (x) ⊂ E u (x) and T x S x = E u Λ (x) for any u-submanifold S x of minimal dimension at x. Moreover, the distribution E u Λ is invariant under the linearizations of the local stable holonomy maps, as well.
We refer the reader to Sect. 2 for the definition of the local stable holonomy maps. In a similar way one defines s-submanifolds of minimal dimension at x ∈ Λ, and a result similar to the above holds for these.
Remark 1. It is easy to see that all vectors in E u (x) that are tangent to the basic set Λ belong to E u Λ (x). However, in general the span of these tangent vectors could be a proper subspace of E u Λ (x) -see Example 2 in Sect. 4 below. The motivation for the introduction of the bundles E u,s Λ comes from [St2] which deals with Dolgopyat type spectral estimates ( [D]) for Axiom A flows on basic sets, where one has to take into account some fine geometric properties of basic sets. As it turns out there, in certain cases when the basic set Λ is 'relatively small', considering the tangent maps dφ t over the whole bundle E u (or E s ) does not say enough about the dynamics of the flow near Λ. So, it appears, restricting dφ t over the smallest possible invariant subbbundle of E u would be beneficial 1 , and it is natural to believe that this might be the case in some other situations, as well, particularly when the dimension of the spaces E u (x) (or E s (x)) is large.
In Sect. 3 below we prove Theorem 1 for flows; for diffeomorphisms the argument is very similar. Some basic definitions are given in Sect. 2, while Sect. 4 contains two examples.

Preliminaries
Let M be a C 1 complete (not necessarily compact) Riemann manifold, and φ t : M −→ M (t ∈ R) a C 1 flow on M . A φ t -invariant closed subset Λ of M is called hyperbolic if Λ contains no fixed points and there exist constants C > 0 and 0 < λ < 1 and a dφ t -invariant decomposition T is the one-dimensional subspace determined by the direction of the flow at x, dφ t (u) ≤ C λ t u for all u ∈ E s (x) and t ≥ 0, and dφ t (u) ≤ C λ −t u for all u ∈ E u (x) and t ≤ 0. Here · is the norm on T x M determined by the Riemann metric on M .
A non-empty compact φ t -invariant hyperbolic subset Λ of M which is not a single closed orbit is called a basic set for φ t if φ t is transitive on Λ and Λ is locally maximal, i.e. there exists an open neighbourhood V of Λ in M such that Λ = ∩ t∈R φ t (V ).
From now on we will assume that Λ is a basic set for the flow φ t . For x ∈ Λ and a sufficiently small ǫ > 0 let Assuming ǫ 1 ∈ (0, ǫ 0 ] is sufficiently small, for any x ∈ Λ and y ∈ Λ∩ W s ǫ 1 (x) the local holonomy map H y x : Λ∩ W u ǫ 1 (x) −→ Λ∩ W u ǫ 0 (y) along stable laminations is well-defined and uniformly Hölder continuous (see e.g. [PSW]). Recall that the map H y x is defined as follows. Given z ∈ Λ ∩ W u ǫ 1 (x), there exist unique z ′ ∈ W s ǫ 1 (z) and y ′ ∈ W u ǫ 0 (y) such that φ t (z ′ ) = y ′ for some t ∈ R, |t| ≤ ǫ 0 . Then we set H y x (z) = y ′ . Under the additional condition that the unstable laminations are C 1 , the maps H y x are C 1 as well (see e.g. Fact (2) on p. 647 in [Ha]). That is, for each z ∈ Λ ∩ W u ǫ 1 (x) the map H y x has a linearization L y x (z) : E u (z) −→ E u (H y x (z)) at z ∈ Λ ∩ W u ǫ 1 (x) and L y x (z) ≤ C for some constant C > 0 independent of x, y and z.
3 Proof of Theorem 1 So, assume that φ t : M −→ M (t ∈ R) is a C 1 flow on M and Λ is a basic set for φ t .
By Whitney's Theorem (see e.g. [M]), the stable holonomy map H y x has a C 1 extension H y be the integers introduced in Sect. 1. Set k = min x∈Λ k x , and fix an arbitrary [KH]). Take t > 0 with this property so large that φ t is expanding on It now follows that k z ′ = k and, since φ t is expanding on W u δ/2 (x), there exists a u-submanifold of minimal dimension at z ′ of size δ/2. The remark in the beginning of the proof now shows that k z = k z ′ = k, and there exists a u-submanifold of minimal dimension at z of size δ/(2C).
Lemma 2. For any x ∈ Λ and any two u-submanifolds S and S ′ of minimal dimension at Choosing a smaller δ > 0 if necessary and using an appropriate submanifold , so the submanifolds S and S ′′ are transversal at x (see e.g. [GP]). Consequently, there exists δ ′ ∈ (0, δ] so that for any t ∈ R and any x ∈ Λ. It remains to show that the map Φ : Λ ∋ x → E u Λ (x) is continuous. Fix for a moment z ∈ Λ and let S z be an u-submanifold of minimal dimension at z. Then for any t ∈ R with a sufficiently small |t|, Next, choosing a sufficiently small δ > 0, for , and therefore Φ(z ′ ) = E u Λ (z ′ ) depends continuously on z ′ ∈ W s ǫ 1 (z)∩Λ. The above and the continuity of the local product near z implies that Φ is continuous at z.

Examples
We use two very well known kind of flows to give: 1) a non-trivial example where k u < dim(E u ), and 2) an example where the span of the tangent vectors in E u (z) to Λ at z form a proper subspace of E u Λ (z) (for any z ∈ Λ). We use open billiard flows for the first example and geodesic flows on hyperbolic manifolds for the second. In higher dimensions both kind of flows should provide a great variety of non-wandering sets with complicated geometric (and metric) structures (see e.g [Ka]), however it appears they would not be easy to investigate. It is quite possible that one could use other models that are easier to describe to get similar examples.
There are various possible ways to define tangent vectors to a subset of a manifold. Here we use the most straightforward one. Given z ∈ Λ, let exp u z : E u (z) −→ W u ǫ 0 (z) and exp s z : It is easy to see that a vector b ∈ E u (z) \ {0} is tangent to Λ at z iff there exists a C 1 curve z(t), 0 ≤ t ≤ a, in W u ǫ (z) for some a > 0 with z(0) = z,ż(0) = b, and z(t) ∈ Λ for arbitrarily small t > 0. Denote by E u (z) the set of all non-zero tangent vectors v ∈ E u (z) to Λ at z. Tangent vectors to Λ in E s (z) and the set E s (z) are defined similarly. Clearly the sets E u,s (z) are non-empty and dφ t -invariant and E u,s (z) ⊂ E u,s (z). However in general the span of E u,s (z) does not coincide with E u,s (z) -see Example 2 below.
Open Problem. Under the assumptions in Sect. 1, assume that k u < dim(E u (z)) (or k s < dim(E s (z))), z ∈ Λ. Does there always exist a C 1 submanifold M ′ of M of positive codimension and an open neighbourhood U of Λ in M such that φ t (U ∩ M ′ ) ⊂ M ′ for all t ∈ R?
One would expect the answer to be negative, however at this stage we do not have examples to prove this. Example 1 below is non-trivial in a certain sense, and in fact might be good enough to answer the above question in the negative, however we do not go that far here.

Open billiard flows
Let K be a subset of R n (n ≥ 2) of the form K = K 1 ∪ K 2 ∪ . . . ∪ K k 0 , where K i are compact strictly convex disjoint domains in R n with C r boundaries Γ i = ∂K i (r ≥ 2) and k 0 ≥ 3. Set Ω = R n \ K and Γ = ∂K. We assume that K satisfies the following no-eclipse condition: for every pair K i , K j of different connected components of K the convex hull of K i ∪ K j has no common points with any other connected component of K. With this condition, the billiard flow φ t defined on the cosphere bundle S * (Ω) in the standard way is called an open billiard flow. It has singularities, however its restriction to the non-wandering set Λ (the set of those x ∈ S * (Ω) such that the trajectory {φ t (x) : t ∈ R} is bounded) has only simple discontinuities at reflection points 2 . Moreover, Λ is compact and φ t is hyperbolic and transitive on Λ. Finally, it follows from [St3] that when the minimal distance between distinct connected components of K is relatively large compared to the maximal sectional curvature of ∂K the open billiard flow on Λ satisfies a certain pinching condition which implies that the (un)stable laminations over Λ are C 1 .
In the following example we will use the natural symbolic coding of the open billiard. Let A be the k 0 × k 0 matrix with entries A(i, j) = 1 if i = j and A(i, i) = 0 for all i, and let Σ A be the set of all sequences η = (η j ) ∞ j=−∞ of integer numbers 1 ≤ η j ≤ k 0 such that η j = η j+1 for all j ∈ Z. Let pr 1 : S * (Ω) −→ Ω be the natural projection. Given ξ ∈ Σ A , let (P j (ξ)) ∞ j=−∞ be the successive reflection points of the unique billiard trajectory in the exterior of K such that P j (ξ) ∈ K ξ j for all j ∈ Z (see e.g. [St1]). Define the map Φ : Σ A −→ Λ ∩ S * ∂K (Ω) by Φ(ξ) = (P 0 (ξ), (P 1 (ξ) − P 0 (ξ))/ P 1 (ξ) − P 0 (ξ) ). Then Φ is a bijection such that Φ is the billiard ball map from boundary to boundary, and σ is the shift map on Σ A .
Example 1. Assume that n = 3 and there exists a plane α such that each of the domains K j is symmetric with respect to α. Setting K ′ = K ∩ α and Ω ′ = Ω ∩ α, it is easy to observe that every billiard trajectory generated by a point in Λ is entirely contained in α. That is, Λ = Λ ′ , where Λ ′ is the non-wandering set for the open billiard flow in Ω ′ . Thus, dim(E u Λ (z)) = 1 < dim(E u (z)) = 2 for any z ∈ Λ. This example is of course trivial, since Λ is contained in the flow-invariant submanifold S * (Ω ′ ) of S * (Ω).
However with a small local perturbation of the boundary ∂K of K we can get a non-trivial example. Choosing standard cartesian coordinates x, y, z in R 3 , we may assume that α is given by the equation z = 0, i.e. α = R 2 × {0}. Let pr 1 : S * (R 3 ) ∼ R 3 × S 2 −→ R 3 be the natural projection, and let C = pr 1 (Λ). We may choose the coordinates x, y in the plane α = {z = 0} so that the line y = 0 is tangent to K ′ 1 and K ′ 2 and K ′ is contained in the half-plane y ≥ 0. Let q 1 ∈ K ′ 1 and q 2 ∈ K ′ 2 be such that [q 1 , q 2 ] is the shortest segment connecting K ′ 1 and K ′ 2 . Take a point q ′ 1 ∈ ∂K ′ 1 close to q 1 and such that the y-coordinate of y ′ 1 is smaller than that of q 1 . Consider the open arc A on ∂K ′ 1 connecting q 1 and q ′ 1 . It is clear that A ∩ C = ∅. Let f : R 3 −→ R 3 be a C 1 (we can make it even C ∞ ) diffeomorphism with f (x) = x for all x outside a small open set U such that q 1 ∈ U and U ∩ ∂K ′ ⊂ A. Then for any q ∈ C the tangent planes T q (∂K) and T q (∂ K) coincide. We can choose f so that K i = f (K i ) = K i for i > 1, K 1 = f (K 1 ) is strictly convex, andν(f (q)) / ∈ α for q ∈ A arbitrarily close to q 1 . Hereν is the outward unit normal field to ∂ K.
Notice that the non-wandering set Λ for the billiard flowφ t in the closure Ω of the exterior of K in R 3 coincides with Λ. Indeed, let Φ : ( Ω) be the coding map for the billiard trajectories in Ω and letz = (q,ξ) ∈ S * ∂ K ( Ω) belong to Λ. Thenz = Φ(ξ) for some ξ ∈ Σ A , so z = Φ(ξ) ∈ Λ ∩ S * ∂K (Ω). Since any reflection point P i (ξ) ∈ C, by the choice of f we have that P j (ξ) are the successive reflection points of a billiard trajectory in Ω and this must be the trajectory determined by Φ(ξ). Thus,z = z, soz ∈ Λ. This argument also shows that Λ ⊂ Λ, so Λ = Λ ⊂ S * (α). Thus, dim(E u Λ (z)) = 1 < dim(E u (z)) for any z ∈ Λ. However, it is clear from the construction that S * (α ∩ Ω) is not invariant with respect to the billiard flowφ t . Moreover, it is not difficult to see that there is no two-dimensional submanifoldα of Ω such that S * (α) is dφ t -invariant and Λ ⊂ S * (α) .
Indeed, assume suchα exists; thenα is a union of billiard trajectories and C = pr 1 (Λ) ⊂α. We will first show thatα = α outside a large disk D containing K ′ . Changing the coordinates x, y in the plane α = {z = 0}, we may assume that the line y = 0 is tangent to K ′ 2 and K ′ 3 and K ′ is contained in the half-plane y ≥ 0. Let p 2 ∈ K ′ 2 and p 3 ∈ K ′ 3 be such that [p 2 , p 3 ] is the shortest segment connecting K ′ 2 and K ′ 3 . Clearly p 2 , p 3 ∈α. Since p 2 is not an isolated point in C, there are points in C ∩ ∂K ′ 2 arbitrarily close to p 2 . All of them are inα, so the curveα ∩ ∂K 2 is tangent to ∂K ′ 2 at p 2 . Similarly, the curveα ∩ ∂K 3 is tangent to ∂K ′ 3 at p 3 . Take point p ′ 2 ∈α ∩ ∂K 2 close to p 2 and p ′ 3 ∈α ∩ ∂K 3 close to p 3 such that the y-coordinates of p ′ 2 and p ′ 3 are smaller than those of p 2 and p 3 , respectively. Consider the arc C 2 onα ∩ ∂K 2 connecting p 2 and p ′ 2 and the arc C 3 onα ∩ ∂K 3 connecting p 3 and p ′ 3 . We have (p 2 , v 2 ) ∈ Λ, where v 2 = (p 3 − p 2 )/ p 3 − p 2 . For any p ∈ C 2 sufficiently close to p 2 and any v ∈ S * p (α) sufficiently close to v 2 , the billiard trajectory γ(p, v) in S * ( Ω) issued from (p, v) is contained in S * (α), so its projection in R 3 is contained inα. Thus, the first reflection point of γ(p, v) belongs to C 3 . That is, for any p ∈ C 2 sufficiently close to p 2 and any v ∈ S * p (α) sufficiently close to v 2 , the straight line segment ray issued from p in the direction of v intersects the curve C 3 . Similarly, for any q ∈ C 3 sufficiently close to p 3 and any w ∈ S * q (α) sufficiently close to −v 2 , the straight line segment ray issued from q in the direction of w intersects the curve C 2 . Replacing the curves C 2 and C 3 by shorter ones (i.e. replacing the points p ′ 2 and p ′ 3 by points on C 2 and C 3 closer to p 2 and p 3 , respectively), we now have that for any p ∈ C 2 and any q ∈ C 3 , the straight line segment [p, q] lies inα. This shows that C 2 and C 3 lie in the same plane, and it is clear from the choice of C 2 and C 3 that this plane is tangent to α at p 2 , so it must coincide with α. Moreover, the union of the segments [p, q] with p ∈ C 2 and any q ∈ C 3 is part of both α andα. From the latter one easily derives that if D is an open disk in α containing K ′ , then α \ D ⊂α. Since Λ is a nowhere dense subset of S * (Ω) (and S * ( Ω), as well), every point q ∈ α \ K can be approximated arbitrarily well by points of the form pr 1 (φ t (x, ξ)), where t > 0 and (x, ξ) ∈ S * (α \ D). This implies α \ K ⊂α, and therefore α \ K =α, which is a contradiction with the definition of K and the perturbation f . Hence there does not exist a two-dimensional submanifold α of Ω such that S * (α) is dφ t -invariant and Λ ⊂ S * (α) .

Geodesic flows on manifolds of negative curvature
Let X be a complete (not necessarily compact) connected Riemann manifold of constant curvature K = −1 and dimension dim(X) = n + 1, n ≥ 1, and let φ t : M = S * (X) −→ M be the geodesic flow on the unit cosphere bundle of X. According to a classical result of Killing and Hopf any such X is a hyperbolic manifold, i.e. X is isometric to H n+1 /Γ, where is the upper half-space in R n+1 with the Poincaré metric ds 2 (x) = 1 x 2 1 (dx 2 1 + . . . + dx 2 1 ) and Γ is a Kleinian group, i.e. a discrete group of isometries (Möbius transformations) acting freely and discontinuously on H n+1 . See e.g. [Ratc] for basic information on hyperbolic manifolds. Given a hyperbolic manifold X = H n+1 /Γ, the limit set L(Γ) is defined as the set of accumulation points of all Γ orbits in ∂H n+1 , the topological closure of ∂H n+1 = {0} × R n including ∞.
Throughout this sub-section we will assume that Γ is torsion-free and finitely generated (then Γ is called geometrically finite) and non-elementary, i.e. L(Γ) is infinite (then L(Γ) is a closed non-empty nowhere dense subset of ∂H n+1 without isolated points; see e.g. Sect. 12.1 in [Ratc]). A geometrically finite Kleinian group with no parabolic elements is called convex cocompact. If X is compact, then Γ is called a cocompact lattice.
The non-wandering set Λ of ϕ t : M −→ M (also known as the convex core of X = H n+1 /Γ) is the image in M of the set of all points of S * (H n+1 ) generating geodesics with end points in L(Γ). When Γ is convex cocompact, the non-wandering set Λ is compact.
Notice that the class of hyperbolic manifolds X = H n+1 /Γ, with Γ a non-elementary convex cocompact Kleinian group, contains all classical and non-classical Schottky manifolds (cf. e.g. Sect. 12.1 in [Ratc]). In this case the (un)stable laminations of the geodesic flow over Λ are always C 1 (in fact C ∞ ).
Example 2. Consider the case n = 2 and let H 3 /Γ be a hyperbolic manifold generated by a non-elementary convex cocompact Kleinian group Γ with an "Apolonian packing" limit set L(Γ) (see e.g. [Su]). Then L(Γ) is a subset of R 2 such that for every x ∈ L(Γ) the set of tangent vectors to L(Γ) at x is one-dimensional, while L(Γ) ∩ U is not contained in an one-dimensional submanifold of R 2 for any open neighbourhood U of x in R 2 . Given z ∈ Λ, the local unstable manifold W u ǫ (z) is given by the projection into H 3 /Γ of the outward normal field to a (part of) a horosphere S in H 3 . The latter are just spheres in H 3 tangent to ∂H 3 or planes in the interior of H 3 parallel to ∂H 3 . Any z ′ ∈ Λ ∩ W u ǫ (z) can be identified with the point x ′ ∈ L(Γ) which is just the limit at +∞ of the geodesic determined by z ′ . It then follows that the set of tangent vectors in E u (z) to Λ at z is one-dimensional, while Λ ∩ U is not contained in an one-dimensional submanifold of W u ǫ (z) for any ǫ > 0, i.e. k u = 2.