Multidimensional expanding maps with singularities: a pedestrian approach

TL;DR

This paper proves the existence of absolutely continuous invariant measures for multidimensional piecewise expanding systems with singularities, using properties of multidimensional BV functions, and studies their statistical behavior.

Contribution

It introduces a new proof technique for invariant measures in complex multidimensional systems with singularities, expanding the theoretical understanding.

Findings

Existence of absolutely continuous invariant measures established.

Analysis of statistical properties of these measures.

Applicable to systems with unbounded derivatives or distortions.

Abstract

I provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses basic properties of multidimensional BV functions.