Lifting Measures to Inducing Schemes

TL;DR

This paper investigates conditions under which invariant measures of certain piecewise continuous maps can be lifted to associated inducing schemes, extending previous results to more complex dynamical systems.

Contribution

It establishes liftability of large entropy measures for maps with inducing schemes, generalizing prior work and applying to multidimensional and cusp maps.

Findings

Large entropy measures can be lifted to inducing schemes under natural conditions.

The results apply to one-dimensional cusp maps and some multidimensional maps.

Connections between inducing schemes and Markov extensions are clarified.

Abstract

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [PS05, PS06]. We show that under some natural assumptions on the inducing schemes - which hold for many known examples - any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [Buz99], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [Bru95] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multimodal maps) and for some multidimensional maps.