STABLE SETS OF CERTAIN NON-UNIFORMLY HYPERBOLIC HORSESHOES HAVE THE EXPECTED DIMENSION

Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes

In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated to Hölder continuous potentials.

Stable Manifolds for Hyperbolic Sets

Multifractal analysis of non-uniformly hyperbolic systems

We prove a multifractal formalism for Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville–Pomeau map.

Constraints On Dynamics Preserving Certain Hyperbolic Sets

We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First if Λ is a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, then Λ is a union of transitive, expanding attractors (or contracting repellers) of codimension 1 for any diffeomorphism such that Λ is hyperbolic. Secondly, if Λ is a nonwandering, locall...

Invariant Pre-foliations for Non-resonant Non-uniformly Hyperbolic Systems

Given an orbit whose linearization has invariant subspaces satisfying some non-resonance conditions in the exponential rates of growth, we prove existence of invariant manifolds tangent to these subspaces. The exponential rates of growth can be understood either in the sense of Lyapunov exponents or in the sense of exponential dichotomies. These manifolds can correspond to “slow manifolds”, whi...