Constructions of subshifts with positive topological entropy dimension

TL;DR

This paper constructs strictly ergodic subshifts with any given entropy dimension between 0 and 1, demonstrating systems with controlled subexponential complexity and regularity, including a weakly mixing variant.

Contribution

It provides a general method to construct subshifts with prescribed entropy dimension and explores their regularity and mixing properties.

Findings

Constructed subshifts with any entropy dimension in (0,1)

Demonstrated regularity in atom sizes and return times

Created a weakly mixing variant system

Abstract

The notion of entropy dimension has been introduced to measure the subexponential complexity of zero entropy systems. In this work we present a general construction of a strictly ergodic subshift of topological entropy dimension $\alpha$ for each $\alpha \in (0,1)$. It is shown that the system satisfies some sort of regularity in the size of atoms and the first return time. Moreover, we modify the construction to obtain a variant system that is weakly mixing.