Geometric rigidity of times-m invariant measures

TL;DR

This paper investigates the rigidity properties of measures invariant under multiplication by integers mod 1 and their behavior under smooth transformations, revealing conditions under which these measures exhibit geometric and algebraic rigidity.

Contribution

It establishes that measures invariant under times-b and times-c transformations exhibit geometric rigidity and mutual singularity under smooth maps, extending previous theorems and clarifying the role of conjugacy classes.

Findings

Measures invariant under multiplication by b are rigid under smooth conjugacy.

Piecewise-analytic maps preserving such measures are piecewise-linear with slopes as rational powers of b.

Invariant measures for different integers c not related to b are mutually singular under smooth maps.

Abstract

Let b be an integer and mu a probability measure on [0,1] which is invariant and ergodic multiplication by b mod 1, and 0<dim(mu)<1. Let f be a diffeomorphism between open subsets of the line. We show that if the measures mu and f(mu) are equivalent on f(E), then f'(x) is a rational power of b at mu-a.e. point x in E. In particular, if g is a piecewise-analytic map preserving mu then there is an open g-invariant set U supporting mu such that g is piecewise-linear on U and has slopes which are rational powers of a. In a similar vein, for mu as above, if c is another integer and b,c are not powers of a common integer, and if nu is in variant for multiplication by c mod 1, then f(mu) and f(nu) are mutually singular for all diffeomorphisms f of class C^2. This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of times-b and times-c is a result not of the structure of the abelian action, but rather of their smooth conjugacy classes: if U,V are maps of the 1-torus which are C^2-conjugate to times-b and times-c, respectively, then they have no common ergodic measures of positive dimension.