# Invariant measures for interval translations and some other piecewise continuous maps

## TL;DR

This paper investigates invariant measures for piecewise continuous maps on manifolds, establishing existence results for interval translation maps and conditions for general maps, linking them to interval exchange maps.

## Contribution

It proves the existence of invariant measures for all interval translation maps and relates these maps to interval exchange maps, extending understanding of their measure-theoretic properties.

## Key findings

- Invariant measures exist for all interval translation maps.
- Interval translation maps are metrically equivalent to interval exchange maps.
- Existence of invariant measures for piecewise maps with wandering discontinuities.

## Abstract

We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided all discontinuity points are wandering.