On the structure of the Birkhoff-irregular set for subshifts of finite type

TL;DR

This paper investigates the complex structure of irregular points in topologically mixing subshifts of finite type, revealing their abundance, intricate disjoint invariant subsets, and full dimensional properties despite measure-zero status.

Contribution

It demonstrates that the irregular set contains uncountably many disjoint invariant subsets with full entropy and dimension, deepening understanding of their complexity.

Findings

Irregular set has full topological entropy and Hausdorff dimension.

Contains uncountably many disjoint dense invariant subsets.

Irregular points exhibit intricate and rich structural properties.

Abstract

We study the set of irregular points for topologically mixing subshifts of finite type. It is well known that despite the irregular set having zero measure for every invariant measure, it has full topological entropy and full Hausdorff dimension. We establish that for these systems the irregular set is not only abundant in terms of its dimensional properties, but also contains uncountably many pairwise disjoint invariant subsets, each of them dense and carrying full topological entropy and Hausdorff dimension. This results deepens our understanding of the complexity of irregular points in dynamical systems, highlighting their intricate structure and suggesting avenues for further explorations in related areas.