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On the ratio ergodic theorem for group actions
Author(s) -
Hochman Michael
Publication year - 2013
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdt022
Subject(s) - ergodic theory , mathematics , abelian group , group (periodic table) , sequence (biology) , combinatorics , polynomial , discrete mathematics , countable set , stationary ergodic process , pure mathematics , invariant measure , mathematical analysis , chemistry , organic chemistry , biology , genetics
We show that the ratio ergodic theorem of Hopf fails in general for measure‐preserving actions of countable amenable groups; in fact, it already fails for the infinite‐rank abelian group⊕ ∞ n = 1Zand many groups of polynomial growth, for instance, the discrete Heisenberg group. More generally, under a technical condition, we show that if the ratio ergodic theorem holds for averages along a sequence of sets { F n } in a group, then there is a finite set E such that { EF n } satisfies the Besicovitch covering property. On the other hand, we prove that in groups with polynomial growth (for which the ratio ergodic theorem sometimes fails) there always exists a sequence of balls along which the ratio ergodic theorem holds if convergence is understood as almost every convergence in density (that is, omitting a sequence of density zero).