Strongly commuting interval maps
Volume 257 / 2022
Fundamenta Mathematicae 257 (2022), 39-68
MSC: 26A21, 37E05, 54H25, 54C10, 54C60.
DOI: 10.4064/fm993-11-2021
Published online: 3 December 2021
Abstract
Maps $f,g\colon I\to I$ are called strongly commuting if $f\circ g^{-1}=g^{-1}\circ f$. We show that surjective, strongly commuting, strictly 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, two strongly commuting, strictly 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.
