A ratio ergodic theorem for multiparameter non-singular actions

Hochman, Michael. "A ratio ergodic theorem for multiparameter non-singular actions." Journal of the European Mathematical Society 012.2 (2010): 365-383. <http://eudml.org/doc/277429>.

@article{Hochman2010,
abstract = {We prove a ratio ergodic theorem for non-singular free $\mathbb \{Z\}^d$ and $\mathbb \{R\}^d$ actions, along balls in an arbitrary norm. Using a Chacon–Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in $\mathbb \{R\}^d$. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group actions, the Besicovitch covering property not only implies the maximal inequality, but is equivalent to it, implying that further generalization may require new methods.},
author = {Hochman, Michael},
journal = {Journal of the European Mathematical Society},
keywords = {group actions; measure preserving transformations; commuting transformations; nonsingular actions; ergodic theorem; maximal inequality; group actions; measure preserving transformations; commuting transformations; nonsingular actions; ergodic theorem},
language = {eng},
number = {2},
pages = {365-383},
publisher = {European Mathematical Society Publishing House},
title = {A ratio ergodic theorem for multiparameter non-singular actions},
url = {http://eudml.org/doc/277429},
volume = {012},
year = {2010},
}

TY - JOUR
AU - Hochman, Michael
TI - A ratio ergodic theorem for multiparameter non-singular actions
JO - Journal of the European Mathematical Society
PY - 2010
PB - European Mathematical Society Publishing House
VL - 012
IS - 2
SP - 365
EP - 383
AB - We prove a ratio ergodic theorem for non-singular free $\mathbb {Z}^d$ and $\mathbb {R}^d$ actions, along balls in an arbitrary norm. Using a Chacon–Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in $\mathbb {R}^d$. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group actions, the Besicovitch covering property not only implies the maximal inequality, but is equivalent to it, implying that further generalization may require new methods.
LA - eng
KW - group actions; measure preserving transformations; commuting transformations; nonsingular actions; ergodic theorem; maximal inequality; group actions; measure preserving transformations; commuting transformations; nonsingular actions; ergodic theorem
UR - http://eudml.org/doc/277429
ER -