Joint ergodicity along generalized linear functions

TL;DR

This paper establishes criteria for joint ergodicity of multiple transformation sequences involving generalized linear functions, including cases with continuous parameters and primes, advancing the understanding of ergodic behavior in dynamical systems.

Contribution

It provides new criteria for joint ergodicity involving generalized linear functions, extending previous results to continuous parameters and prime sequences.

Findings

Criteria for joint ergodicity of sequences with generalized linear functions.

Extension of ergodicity criteria to transformations depending on continuous parameters.

Conditions for joint ergodicity along prime sequences.

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 of a continuous parameter, as well as a condition of joint ergodicity of sequences $T_{i}^{\phi_{i}(n)}$ along primes.