# Geometric rigidity of $×m$ invariant measures

### Michael Hochman

The Hebrew University of Jerusalem, Israel

## Abstract

Let $μ$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_{a}(x)=axmod1$, and $0<dimμ<1$. Let $f$ be a local diffeomorphism on some open set. We show that if $E⊆R$ and $(fμ)∣_{E}∼μ∣_{E}$, then $f_{′}(x)∈{±a_{r}:r∈Q}$ at $μ$-a.e. point $x∈f_{−1}E$. In particular, if $g$ is a piecewise-analytic map preserving $μ$ then there is an open $g$-invariant set $U$ containing supp $μ$ such that $g∣_{U}$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $μ$ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $ν$ is a $T_{b}$-invariant measure, then $fμ⊥ν$ 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 $R/Z$ 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 $×m$ invariant measures. J. Eur. Math. Soc. 14 (2012), no. 5, pp. 1539–1563

DOI 10.4171/JEMS/340