Symbolic dynamics for one dimensional maps with nonuniform expansion

Abstract

Given a piecewise C 1+β map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, for each χ > 0 we construct a finite-to-one Hölder continuous map from a countable topological Markov shift to the natural extension of the interval map, that codes the lifts of all invariant probability measures as above with Lyapunov exponent greater than χ almost everywhere.

Invariant measures for interval maps without Lyapunov exponents

“…Recently Dobbs [Dob14], [Dob15] developed the Pesin theory for noninvertible interval maps with Lorenz-like singularities and non-flat critical points. Lima [Lim20] constructs a symbolic extension for these maps that code the measures with positive Lyapunov exponents.…”

“…The negation of item (3) in Theorem 1 is considered in several works as a regularity condition to study ergodic invariant measures. In [Lim20], Lima studied measures satisfying this condition for interval maps with critical points and discontinuities, he called measures satisfying this condition f −adapted. By the Birkhoff ergodic theorem, if log(dist(•, S(f ))) ∈ L 1 (µ), then for an ergodic invariant measure µ, we have…”

“…The measure μP does not satisfy the non-degenerated condition. For more results related to this condition see [Lim20] and references therein.…”

Symbolic dynamics for nonuniformly hyperbolic maps with singularities in high dimension

“…This unified framework allows us to cover many classical examples of nonuniform hyperbolic dynamics, such as geodesic flows in closed manifolds, billiard maps, and Viana maps. Also, this wider framework includes the four works cited above, in some cases providing even better results (in comparison to [Lim20]). Not only this: using the recent work of Ben Ovadia [BO20], we are able to identify the points coded by recurrent sequences.…”

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

“…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.…”