Sequence entropy tuples and mean sensitive tuples

TL;DR

This paper explores the relationship between sequence entropy tuples and mean sensitive tuples in dynamical systems, establishing their equivalence under certain conditions in both topological and measure-theoretical contexts.

Contribution

It characterizes sequence entropy tuples via mean sensitive tuples and proves their equivalence in ergodic measure-preserving systems and specific minimal systems.

Findings

In ergodic systems, sequence entropy, mean sensitive, and sensitive in the mean tuples coincide.

An example demonstrates the necessity of ergodicity for the equivalence.

In certain minimal systems, mean sensitive tuples are identical to sequence entropy tuples.

Abstract

Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the $\mu$-sequence entropy tuple, the $\mu$-mean sensitive tuple and the $\mu$-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.