Symbolic dynamics for one dimensional maps with nonuniform expansion

TL;DR

This paper develops a symbolic coding framework for invariant measures of one-dimensional maps with nonuniform expansion, even in the presence of critical points and discontinuities, under certain conditions.

Contribution

It introduces a method to code invariant measures with nonuniform expansion for piecewise $C^{1+eta}$ maps, including those with critical points and discontinuities, in the natural extension.

Findings

Constructed symbolic models for nonuniformly expanding measures

Applicable to maps with critical points and discontinuities

Provides a coding for measures that avoid critical regions exponentially fast

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, we code the lift of these measures in the natural extension of the map.