  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.
