Geometric rigidity of $\times m$ invariant measures

Michael Hochman

Journal of the European Mathematical Society, Tome 14 (2012) no. 5, pp. 1539-1563

Résumé

Let μ be a probability measure on [0,1] which is invariant and ergodic for Ta​(x)=axmod1, and 01. Let f be a local diffeomorphism on some open set. We show that if E⊆R and (fμ)∣E​∼μ∣E​, then f′(x)∈{±ar:r∈Q} at μ-a.e. point x∈f−1E. 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 Tb​-invariant measure, then fμ⊥ν for all local diffeomorphisms f of class C2. This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of Ta​,Tb​ 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 C2-conjugate to Ta​,Tb​ then they have no common measures of positive dimension which are ergodic for both.

Détail

DOI : 10.4171/jems/340

Classification : 37-XX, 28-XX, 00-XX
Keywords: Measure rigidity, invariant measure, interval map, fractal geometry, geometric measure theory, scenery flow