On a class of sofic affine invariant subsets of the 2-torus related to an Erdős problem
Résumé
Let 1 β 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.