Box dimensions of $(\times m, \times n)$-invariant sets

TL;DR

This paper investigates the box dimensions of invariant sets under specific toral endomorphisms, especially focusing on cases where the underlying symbolic dynamics are not topologically mixing or sofic, providing bounds and formulas for these dimensions.

Contribution

It extends the understanding of box dimensions for invariant sets beyond well-understood mixing cases, offering bounds and explicit formulas for non-sofic and non-mixing subshifts.

Findings

Upper bounds for box dimensions are established for all subshifts.

The upper bound is exact for certain coded subshifts with freely concatenated words.

Examples show the bounds can be strict, and formulas are provided for non-transitive sofic cases.

Abstract

We study the box dimensions of sets invariant under the toral endomorphism $(x, y) \mapsto (m x \text{ mod } 1, \, n y \text{ mod } 1)$ for integers $n>m \geq 2$. The basic examples of such sets are Bedford-McMullen carpets and, more generally, invariant sets are modelled by subshifts on the associated symbolic space. When this subshift is topologically mixing and sofic the situation is well-understood by results of Kenyon and Peres. Moreover, other work of Kenyon and Peres shows that the Hausdorff dimension is generally given by a variational principle. Therefore, our work is focused on the box dimensions in the case where the underlying shift is not topologically mixing and sofic. We establish straightforward upper and lower bounds for the box dimensions in terms of entropy which hold for all subshifts and show that the upper bound is the correct value for coded subshifts whose entropy can be realised by words which can be freely concatenated, which includes many well-known families such as $\beta$-shifts, (generalised) $S$-gap shifts, and transitive sofic shifts. We also provide examples of transitive coded subshifts where the general upper bound fails and the box dimension is actually given by the general lower bound. In the non-transitive sofic setting, we provide a formula for the box dimensions which is often intermediate between the general lower and upper bounds.