Explosion of smoothness for conjugacies between multimodal maps

TL;DR

This paper demonstrates that under certain conditions, a topological conjugacy between smooth multimodal maps becomes globally smooth, especially near expanding sets and boundary points, revealing a 'smoothness explosion' phenomenon.

Contribution

It establishes that local smoothness of conjugacies implies global smoothness in the basin of renormalization intervals for multimodal maps.

Findings

Conjugacies are smooth in the basin of a renormalization interval.

Smoothness at a boundary point extends to the entire interval.

The result applies to $C^r$ unimodal maps with boundary smoothness.

Abstract

Let $f$ and $g$ be smooth multimodal maps with no periodic attractors and no neutral points. If a topological conjugacy $h$ between $f$ and $g$ is $C^{1}$ at a point in the nearby expanding set of $f$, then $h$ is a smooth diffeomorphism in the basin of attraction of a renormalization interval of $f$. In particular, if $f:I \to I$ and $g:J \to J$ are $C^r$ unimodal maps and $h$ is $C^{1}$ at a boundary of $I$ then $h$ is $C^r$ in $I$.