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On a class of sofic affine invariant subsets of the 2-torus related to an Erdős problem

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Abstract

Let 1 < β < 2 be a real number and G be the closed projection on the 2-torus of the (modified) Rademacher graph in base β. The smallest compact containing G and left invariant by the diagonal endomorphism \({(x,y)\mapsto(2x,\beta y)}\) (mod 1) is denoted by K. For β a simple Parry number of PV-type, K is proved to be a sofic affine invariant set with a fractal geometry closed to the one of G. When β is the golden number, we prove the uniqueness of the measure with full Hausdorff dimension on K.

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  1. Université de Provence, 3 Pl. V. Hugo, 13331, Marseille Cedex 03, France

    Eric Olivier

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Correspondence to Eric Olivier.

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Communicated by Klaus Schmidt.

To the memory of Gérard Rauzy.

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Olivier, E. On a class of sofic affine invariant subsets of the 2-torus related to an Erdős problem. Monatsh Math 165, 447–497 (2012). https://doi.org/10.1007/s00605-011-0296-2

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