Distributional chaos in multifractal analysis, recurrence and transitivity

TL;DR

This paper investigates distributional chaos (DC1) in irregular and level sets of dynamical systems with the specification property, showing the existence of uncountable DC1-scrambled subsets in these complex sets.

Contribution

It demonstrates the presence of uncountable DC1-scrambled subsets in irregular and level sets for systems with the specification property, linking chaos with multifractal analysis.

Findings

Uncountable DC1-scrambled subsets exist in irregular sets.

Uncountable DC1-scrambled subsets are present in various recurrent level sets.

The main technique involves constructing DC1-scrambled sets in saturated sets.

Abstract

There are lots of results to study dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure) etc.. However, it is unknown from the viewpoint of chaos. There are lots of results on the relationship of positive topological entropy and various chaos but it is known that positive topological entropy does not imply a strong version of chaos called DC1 so that it is non-trivial to study DC1 on irregular sets and level sets. In this paper we will show that for dynamical system with specification property, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. On the other hand, we also prove that several recurrent levels of points with different recurrent frequency all have uncountable DC1-scrambled subsets. The main technique established to prove above results is that there exists uncountable DC1-scrambled subset in saturated sets.