# Geometric rigidity of $\times m$ invariant measures

### Michael Hochman

The Hebrew University of Jerusalem, Israel

## Abstract

Let $\mu$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_{a}(x)=ax\bmod1$, 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 $f'(x)\in\{\pm a^{r}\,:\, r\in\mathbb{Q}\}$ 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.

## Cite this article

Michael Hochman, Geometric rigidity of $\times m$ invariant measures. J. Eur. Math. Soc. 14 (2012), no. 5, pp. 1539–1563

DOI 10.4171/JEMS/340