Geometric rigidity of invariant measures

  • Michael Hochman

    The Hebrew University of Jerusalem, Israel


Let be a probability measure on which is invariant and ergodic for , and . Let be a local diffeomorphism on some open set. We show that if and , then at -a.e. point . In particular, if is a piecewise-analytic map preserving then there is an open -invariant set containing supp such that is piecewise-linear with slopes which are rational powers of . In a similar vein, for as above, if is another integer and are not powers of a common integer, and if is a -invariant measure, then for all local diffeomorphisms of class . This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of is a result not of the structure of the abelian action, but rather of their smooth conjugacy classes: if are maps of which are -conjugate to then they have no common measures of positive dimension which are ergodic for both.

A correction to this paper is available.

Cite this article

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

DOI 10.4171/JEMS/340