Strongly commuting interval maps

TL;DR

This paper investigates strongly commuting piecewise monotone interval maps, showing they can be decomposed into invariant intervals with specific properties, and proving they share a common fixed point, impacting the understanding of their dynamical behavior.

Contribution

It provides a decomposition theorem for strongly commuting maps and establishes the existence of a common fixed point, advancing the understanding of their structure and dynamics.

Findings

Decomposition of strongly commuting maps into invariant intervals.

Existence of a common fixed point for such maps.

Implications for dynamical properties on inverse limit spaces.

Abstract

Maps $f,g\colon I\to I$ are called strongly commuting if $f\circ g^{-1}=g^{-1}\circ f$. We show that strongly commuting, piecewise monotone maps $f,g$ can be decomposed into a finite number of invariant intervals (or period 2 intervals) on which $f,g$ are either both open maps, or at least one of them is monotone. As a consequence, we show that strongly commuting piecewise monotone interval maps have a common fixed point. Results of the paper also have implications in understanding dynamical properties of certain maps on inverse limit spaces.