Entropy of AT(n) systems

TL;DR

This paper proves that ergodic AT(n) systems have zero entropy, demonstrates that Bernoulli shifts are not AT(n), and provides an example of zero entropy systems lacking AT(n) property.

Contribution

It establishes a connection between AT(n) property and zero entropy, and clarifies the limitations of AT(n) systems in ergodic theory.

Findings

Ergodic AT(n) systems have zero entropy.

Bernoulli shifts are not AT(n).

Existence of zero entropy systems without AT(n) property.

Abstract

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is AT$(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not AT($n$). We also give an example of a transformation which has zero entropy but does not have property AT($n$), for any integer $n\geq 1$.