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\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.