Geometry of expanding absolutely continuous invariant measures and the liftability problem

TL;DR

This paper establishes that for many manifold maps, having a Gibbs-Markov-Young structure is both necessary and sufficient for the existence of an absolutely continuous invariant measure with positive Lyapunov exponents.

Contribution

It proves a fundamental equivalence between Gibbs-Markov-Young structures and absolutely continuous invariant measures with positive Lyapunov exponents for a broad class of maps.

Findings

Gibbs-Markov-Young structure characterizes absolutely continuous measures

Positive Lyapunov exponents are linked to the structure's existence

Necessary and sufficient conditions are identified for invariant measures

Abstract

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs-Markov-Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.