On isotopy and unimodal inverse limit spaces

TL;DR

This paper proves that all self-homeomorphisms of certain inverse limit spaces of tent maps are isotopic to powers of the shift map, revealing a fundamental structural property of these dynamical systems.

Contribution

It establishes that every self-homeomorphism on the inverse limit space of tent maps with slopes in (√2, 2] is isotopic to a shift power, a novel structural insight.

Findings

All self-homeomorphisms are isotopic to shift powers

Inverse limit spaces of tent maps have a rigid isotopy class structure

The result applies to tent maps with slopes in (√2, 2]

Abstract

We prove that every self-homeomorphism $h : K_s \to K_s$ on the inverse limit space $K_s$ of tent map $T_s$ with slope $s \in (\sqrt 2, 2]$ is isotopic to a power of the shift-homeomorphism $\sigma^R : K_s \to K_s$.