Hausdorff dimension of specification for the $(\alpha,\beta)$-shifts

TL;DR

This paper investigates the Hausdorff dimension of the set of parameters for which $( ext{ extalpha}, ext{eta})$-shifts exhibit the specification property, revealing it has full dimension 2 in the parameter space.

Contribution

It extends previous results by showing the Hausdorff dimension of the parameter set with specification is 2, using a novel approach with intersections of thick Cantor sets.

Findings

Hausdorff dimension of the parameter set is 2

The set of $eta$-shifts with specification has dimension 1

The set of $(-eta)$-shifts with specification has dimension 1

Abstract

Specification is an important concept in dynamical systems introduced by Bowen. Schmeling proved that the set of $\beta>1$ such that the corresponding $\beta$-shift has specification is of Hausdorff dimension $1$. Hu et al. proved that the set of $\beta>1$ such that the corresponding $(-\beta)$-shift has specification is of Hausdorff dimension $1$. We show that the set of $(\alpha,\beta)\in[0,1)\times(1,\infty)$ such that the corresponding $(\alpha,\beta)$-shift has specification is of Hausdorff dimension $2$. A new difficulty is a simultaneous control of two critical symbol sequences that determine the ambient shift space. We achieve this by taking intersections of two thick Cantor sets in parameter space.