Some Generic Properties of Processes

TL;DR

This paper explores generic properties of processes generated by ergodic measure-preserving transformations, revealing typical behaviors such as non-Rosenblatt mixing and characteristics of K-automorphisms, with extensions to broader group actions.

Contribution

It characterizes the typical properties of processes derived from ergodic transformations, including mixing and Bernoulli properties, using a complete metric on partitions.

Findings

Generic process is not Rosenblatt mixing

Generic K-automorphism is not Bernoulli

Extensions to amenable group actions discussed

Abstract

For a given ergodic measure preserving transformation T of a standard measure space each finite labelled partition defines an ergodic stationary process. There is a complete metric on the space of partitions which is separable. Various generic properties of these processes will be given. For example: 1. The generic partition defines a process that is not Rosenblatt mixing. 2. If T is a K-automorphism that is not Bernoulli then the generic partition is also K but not Bernoulli. Extensions to the relative setting and to actions of amenable groups will also be discussed.