Geometric rigidity of $\times m$ invariant measures

Hochman, Michael. "Geometric rigidity of $\times m$ invariant measures." Journal of the European Mathematical Society 014.5 (2012): 1539-1563. <http://eudml.org/doc/277315>.

@article{Hochman2012,
abstract = {Let $\mu $ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_a(x)=ax \left. \texttt \{mod\} \left. 1\right.\right.$, and $0<\texttt \{dim\}\left. \mu < 1\right.$. Let $f$ be a local diffeomorphism on some open set. We show that if $E\subseteq \mathbb \{R\}$ and $(f\mu )\left|_E \sim \mu \right|_E$, then $f^\{\prime \}(x)\in \left\lbrace \pm a^r : r \in \mathbb \{Q\}\right\rbrace $ at $\mu $-a.e. point $x\in f^\{-1\}E$. In particular, if $g$ is a piecewise-analytic map preserving $\mu $ then there is an open $g$-invariant set $U$ containing supp $\mu $ such that $g\left|_U\right.$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $\mu $ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $\nu $ is a $T_b$-invariant measure, then $f\mu \bot \nu $ for all local diffeomorphisms $f$ of class $C^2$. This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of $T_a,T_b$ is a result not of the structure of the abelian action, but rather of their smooth conjugacy classes: if $U,V$ are maps of $\mathbb \{R\}/\mathbb \{Z\}$ which are $C^2$-conjugate to $T_a,T_b$ then they have no common measures of positive dimension which are ergodic for both.},
author = {Hochman, Michael},
journal = {Journal of the European Mathematical Society},
keywords = {measure rigidity; invariant measure; interval map; fractal geometry; geometric measure theory; scenery flow; measure rigidity; invariant measure; interval map; fractal geometry; geometric measure theory; scenery flow},
language = {eng},
number = {5},
pages = {1539-1563},
publisher = {European Mathematical Society Publishing House},
title = {Geometric rigidity of $\times m$ invariant measures},
url = {http://eudml.org/doc/277315},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Hochman, Michael
TI - Geometric rigidity of $\times m$ invariant measures
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 5
SP - 1539
EP - 1563
AB - Let $\mu $ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_a(x)=ax \left. \texttt {mod} \left. 1\right.\right.$, and $0<\texttt {dim}\left. \mu < 1\right.$. Let $f$ be a local diffeomorphism on some open set. We show that if $E\subseteq \mathbb {R}$ and $(f\mu )\left|_E \sim \mu \right|_E$, then $f^{\prime }(x)\in \left\lbrace \pm a^r : r \in \mathbb {Q}\right\rbrace $ at $\mu $-a.e. point $x\in f^{-1}E$. In particular, if $g$ is a piecewise-analytic map preserving $\mu $ then there is an open $g$-invariant set $U$ containing supp $\mu $ such that $g\left|_U\right.$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $\mu $ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $\nu $ is a $T_b$-invariant measure, then $f\mu \bot \nu $ for all local diffeomorphisms $f$ of class $C^2$. This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of $T_a,T_b$ is a result not of the structure of the abelian action, but rather of their smooth conjugacy classes: if $U,V$ are maps of $\mathbb {R}/\mathbb {Z}$ which are $C^2$-conjugate to $T_a,T_b$ then they have no common measures of positive dimension which are ergodic for both.
LA - eng
KW - measure rigidity; invariant measure; interval map; fractal geometry; geometric measure theory; scenery flow; measure rigidity; invariant measure; interval map; fractal geometry; geometric measure theory; scenery flow
UR - http://eudml.org/doc/277315
ER -