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

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.

arXiv comment10 pages, no figure