Symbolic dynamics for one dimensional maps with nonuniform expansion

Abstract

Given a piecewise $C^{1+\beta }$ 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 $\chi >0$ 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.