On the ratio ergodic theorem for group actions

TL;DR

This paper investigates the conditions under which the ratio ergodic theorem (RET) holds for various group actions, revealing limitations in certain groups and establishing cases where RET can hold with density-based convergence.

Contribution

It characterizes when the RET applies to group actions, showing that for many groups, standard sequences like balls do not satisfy RET, but in polynomial growth groups, RET can hold with density convergence.

Findings

RET fails for free abelian groups of infinite rank with standard sequences

RET does not hold along balls in groups like the Heisenberg group

RET holds in polynomial growth groups with density-based convergence

Abstract

We study the ratio ergodic theorem (RET) of Hopf for group actions. Under a certain technical condition, if a sequence of sets {F_n} in a group satisfy the RET, then there is a finite set E such that {EF_n} satisfies the Besicovitch covering property. Consequently for free abelian groups of infinite rank there is no sequence F_n along which the RET holds, and in many finitely generated groups, including the discrete Heisenberg group and the free group on d>1 generators, there is no (sub)sequence of balls, in the standard generators, along which the RET holds. On the other hand, in groups with polynomial growth (including the Heisenberg group, to which our negative results apply) there always exists a sequence of balls along which the RET holds if convergence is understood as a.e. convergence in density (i.e. omitting a sequence of density zero).