Highly irregular orbits for subshifts of finite type: large intersections and emergence

TL;DR

This paper investigates the complexity of irregular orbits in subshifts of finite type, demonstrating that points with high pointwise emergence form large, rich sets with full entropy, dimension, and pressure, extending previous work on residual sets.

Contribution

It generalizes the concept of high emergence to subshifts of finite type, showing these sets have full topological entropy, Hausdorff dimension, and pressure, and belong to large intersection classes.

Findings

Set of points with high pointwise emergence has full topological entropy.

Such sets have full Hausdorff dimension.

They also have full topological pressure for any Hölder potential.

Abstract

In their recent paper [KNS2019], the first author, S. Kiriki, and T. Soma introduced a concept of pointwise emergence to measure the complexity of irregular orbits. They constructed a residual subset of the full shift with high pointwise emergence. In this paper we consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. We show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any H\"older continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property. This is a natural generalization of [FP2011] to pointwise emergence and Carath\'eodory dimension.