Smooth symmetries of $\times a$-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 $\log f^{\prime }(x)/\log a\in \mathbb{Q}$ 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.

14 pages. v2: some corrections and tweeks