Expanding Measures

TL;DR

This paper extends a classical one-dimensional theorem to higher dimensions, proving the existence of absolutely continuous invariant measures for certain transformations with critical or singular regions.

Contribution

It generalizes Keller's theorem to arbitrary dimensions and constructs an induced Markov map to analyze statistical properties of the transformation.

Findings

Existence of absolutely continuous invariant measures under mild conditions.

Construction of an induced Markov map with bounded hyperbolic times.

Application of the induced map to study decay of correlations.

Abstract

We prove that any C^{1+} transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller for maps of the interval with negative Schwarzian derivative. We also show how to construct an induced Markov map F such that every expanding probability of the initial transformation lifts to an invariant probability of F. The induced time is bounded at each point by the corresponding first hyperbolic time (the first time the dynamics exhibits hyperbolic behavior). In particular, F may be used to study decay of correlations and others statistical properties of the initial map, relative to any expanding probability.