JournalsjemsVol. 14, No. 5pp. 1539–1563

Geometric rigidity of ×m\times m invariant measures

  • Michael Hochman

    The Hebrew University of Jerusalem, Israel
Geometric rigidity of $\times m$  invariant measures cover
Download PDF

Abstract

Let μ\mu be a probability measure on [0,1][0,1] which is invariant and ergodic for Ta(x)=axmod1T_{a}(x)=ax\bmod1, and 0<dimμ<10<\dim\mu<1. Let ff be a local diffeomorphism on some open set. We show that if ERE\subseteq\mathbb{R} and (fμ)EμE(f\mu)|_{E}\sim\mu|_{E}, then f(x){±ar:rQ}f'(x)\in\{\pm a^{r}\,:\, r\in\mathbb{Q}\} at μ\mu-a.e. point xf1Ex\in f^{-1}E. In particular, if gg is a piecewise-analytic map preserving μ\mu then there is an open gg-invariant set UU containing supp μ\mu such that gUg|_{U} is piecewise-linear with slopes which are rational powers of aa. In a similar vein, for μ\mu as above, if bb is another integer and a,ba,b are not powers of a common integer, and if ν\nu is a TbT_{b}-invariant measure, then fμνf\mu\perp\nu for all local diffeomorphisms ff of class C2C^{2}. This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of Ta,TbT_{a},T_{b} is a result not of the structure of the abelian action, but rather of their smooth conjugacy classes: if U,VU,V are maps of R/Z\mathbb{R}/\mathbb{Z} which are C2C^{2}-conjugate to Ta,TbT_{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\times m invariant measures. J. Eur. Math. Soc. 14 (2012), no. 5, pp. 1539–1563

DOI 10.4171/JEMS/340