Measure-theoretic sensitivity via finite partitions

TL;DR

This paper introduces measure-theoretic n-sensitivity using finite partitions and characterizes ergodic systems with maximal pattern entropy log n as exactly those that are n-sensitive but not (n+1)-sensitive.

Contribution

It defines measure-theoretic n-sensitivity via finite partitions and links it to maximal pattern entropy in ergodic systems, providing a new characterization.

Findings

Ergodic systems are n-sensitive but not (n+1)-sensitive iff their maximal pattern entropy is log n.

Introduces measure-theoretic n-sensitivity concept for dynamical systems.

Establishes a precise relationship between sensitivity levels and pattern entropy.

Abstract

For every positive integer $n\geq 2$, we introduce the concept of measure-theoretic $n$-sensitivity for measure-theoretic dynamical systems via finite measurable partitions, and show that an ergodic system is measure-theoretically $n$-sensitive but not $(n+1)$-sensitive if and only if its maximal pattern entropy is $\log n$.