Smooth symmetries of $\times a$-invariant sets

TL;DR

This paper investigates smooth functions on sets invariant under multiplication by a and demonstrates that their derivatives' logarithms are rational multiples of log a at many points, using advanced dynamical systems techniques.

Contribution

It introduces a new topological scenery flow method and extends previous results to zero-entropy cases for invariant sets.

Findings

Logarithm of the derivative divided by log a is rational at many points.

The method combines scenery flow and equidistribution techniques.

Extends results to zero-entropy invariant sets.

Abstract

We study the smooth self-maps $f$ of $\times a$-invariant sets $X\subseteq[0,1]$. Under various assumptions we show that this forces $\log f'(x)/\log a\in\mathbb{Q}$ at many points in $X$. Our method combines scenery flow methods and equidistribution results in the positive entropy case, where we improve previous work of the author and Shmerkin, with a new topological variant of the scenery flow which applies in the zero-entropy case.