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.
Cite this article
Michael Hochman, Geometric rigidity of invariant measures. J. Eur. Math. Soc. 14 (2012), no. 5, pp. 1539–1563DOI 10.4171/JEMS/340