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Abstract
We study the box dimensions of sets invariant under the toral endomorphism (x, y) 7→ (mx mod 1, ny mod 1) for integers n > m ≥ 2. The basic examples of such sets are BedfordMcMullen 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 wellunderstood by results of Kenyon and Peres. Moreover, other work of Kenyona nd 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 wellknown families such as βshifts,(generalised) Sgap shifts, and topologically transitive sofic shifts. We also provide examples of topologically mixing coded subshifts where the general upper bound fails and the box dimension is actually given by the general lower bound. In the nontransitive sofic setting, we provide a formula for the box dimensions which is often intermediate between the general lower and upper bounds.
Original language  English 

Pages (fromto)  23412367 
Number of pages  27 
Journal  Indiana University Mathematics Journal 
Volume  72 
Issue number  6 
DOIs  
Publication status  Published  1 Dec 2023 
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Dive into the research topics of 'Box dimensions of (×m,×n)invariant sets'. Together they form a unique fingerprint.Projects
 2 Finished

New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
1/09/19 → 31/01/23
Project: Standard

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
1/02/18 → 11/06/21
Project: Standard