On the topological conjugacy problem for interval maps

TL;DR

This paper introduces a novel inverse approach to determine topological conjugacy of interval maps by analyzing their branch relations, offering a new framework for an open problem in ergodic theory with implications for spectral properties.

Contribution

It presents a symmetry-breaking framework for topological conjugacy of interval maps, shifting focus from map equations to branch relations, addressing a longstanding open problem.

Findings

Framework for topological conjugacy based on branch relations

Implications for spectrum and eigenfunctions of Perron-Frobenius operator

New perspective on ergodic theory problems

Abstract

We propose an inverse approach for dealing with interval maps based on the manner whereby their branches are related (folding property), instead of addressing the map equations as a whole. As a main result, we provide a symmetry-breaking framework for determining topological conjugacy of interval maps, a well-known open problem in ergodic theory. Implications thereof for the spectrum and eigenfunctions of the Perron-Frobenius operator are also discussed.