论文信息 - On the ratio ergodic theorem for group actions

On the ratio ergodic theorem for group actions

We study the ratio ergodic theorem (RET) of Hopf for group actions. Under a certain technical condition, if a sequence of sets {Fn} in a group satisfy the RET, then there is a finite set E such that {EFn} satisfies the Besicovitch covering property. Consequently for the abelian group G = ⊕n=1Z there is no sequence Fn ⊆ G along which the RET holds, and in many finitely generated groups, including the discrete Heisenberg group and the free group on ≥ 2 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).

Michael Hochman

[1] M. Gromov. Groups of polynomial growth and expanding maps , 1981 .

[2] E. Lindenstrauss. Invariant measures and arithmetic quantum unique ergodicity , 2006 .

[3] Amos Nevo,et al. Chapter 13 - Pointwise Ergodic Theorems for Actions of Groups , 2006 .

[5] J. Feldman. A ratio ergodic theorem for commuting, conservative, invertible transformations with quasi-invariant measure summed over symmetric hypercubes , 2007, Ergodic Theory and Dynamical Systems.

[6] M. E. Becker. A ratio ergodic theorem for groups of measure-preserving transformations , 1983 .

[7] M. Hochman. A ratio ergodic theorem for multiparameter non-singular actions , 2009, 0901.3605.

[8] Hyman Bass,et al. The Degree of Polynomial Growth of Finitely Generated Nilpotent Groups , 1972 .

[9] A. Nevo,et al. A horospherical ratio ergodic theorem for actions of free groups , 2012, 1207.3569.