Joint ergodicity along generalized linear functions

Abstract

A criterion of joint ergodicity of several sequences of transformations of a probability measure space X of the form T-i(phi i(n)) is given for the case where T-i are commuting measure-preserving transformations of X and phi(i) are integer-valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending on a continuous parameter, as well as a condition of joint ergodicity of sequences T-i(phi i(n)) along primes.