Invariant measures for piesewise fractional linear maps

TL;DR

This paper investigates invariant measures for piecewise fractional linear maps with three branches, exploring the existence of common measures for related maps and introducing new types of invariant measures.

Contribution

It introduces conditions for common invariant measures between related maps and presents novel invariant measures for piecewise fractional linear maps.

Findings

Conditions for common invariant measures identified

New types of invariant measures constructed

Analysis of map relations and measure invariance

Abstract

The first part deals with piecewise fractional linear maps with three branches. Given a map $T$ a map $S$ is called a related map if some branches of $T$ are replaced by a 'flipped' branch, namely a branch of $1-T$. The main question is if $T$ and $S$ have a common invariant measure. The short second part presents invariant measures of a new type.