Geometric rigidity of $\times m$ invariant measures

Abstract

Let $\mu$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_a(x)=ax\mathrm{mod}1$, and $0<\dim \mu <1$. Let $f$ be a local diffeomorphism on some open set. We show that if $E\subseteq \mathbb{R}$ and $(f\mu ){|}_E\sim \mu {|}_E$, then $f^{\prime }(x)\in \{\pm a^r:r\in \mathbb{Q}\}$ at $\mu$-a.e. point $x\in f^{-1}E$. In particular, if $g$ is a piecewise-analytic map preserving $\mu$ then there is an open $g$-invariant set $U$ containing supp $\mu$ such that $g{|}_U$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $\mu$ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $\nu$ is a $T_b$-invariant measure, then $f\mu \perp \nu$ for all local diffeomorphisms $f$ of class $C^2$. This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of $T_a,T_b$ is a result not of the structure of the abelian action, but rather of their smooth conjugacy classes: if $U,V$ are maps of $\mathbb{R}/\mathbb{Z}$ which are $C^2$-conjugate to $T_a,T_b$ then they have no common measures of positive dimension which are ergodic for both.

A correction to this paper is available.