Invariant measures for piecewise continuous maps

TL;DR

This paper proves the existence of invariant measures for a broad class of piecewise continuous interval maps and establishes a semi-conjugacy to interval exchange transformations under certain conditions.

Contribution

It introduces new results on invariant measures and semi-conjugacy for piecewise continuous maps without connections, expanding understanding of their dynamical properties.

Findings

Invariant Borel probability measures exist for all connection-free piecewise continuous maps.

Injective maps with no connections or periodic orbits are semi-conjugate to interval exchange transformations.

Provides a framework linking piecewise continuous maps to well-studied interval exchanges.

Abstract

We say that $f:[0,1]\to [0,1]$ is a {\it piecewise continuous interval map} if there exists a partition $0=x_0<x_1<\cdots<x_{d}<x_{d+1}=1$ of $[0,1]$ such that $f\vert_{(x_{i-1},x_i)}$ is continuous and the lateral limits $w_0^+=\lim_{x\to 0^+} f(x)$, $w_{d+1}^-=\lim_{x\to 1^-} f(x)$, \mbox{$w_i^{-}=\lim_{x\to x_i^-} f(x)$} and $w_i^{+}=\lim_{x\to x_i^+} f(x)$ exist for each $i$. We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. We also prove that every injective piecewise continuous interval map with no connections and no periodic orbits is topologically semi-conjugate to an interval exchange transformation.