On a class of sofic affine invariant subsets of the 2-torus related to an Erdős problem (Q766213)

The author considers the smallest compact set \(K\) which contains the closed projection \(G\) on the \(2\)-torus of the modified Rademacher graph in base \(\beta\) (for some \(1<\beta<2\)) such that \(K\) is invariant under the diagonal endomorphism \(T\) which maps the pair \((x,y)\) to the pair \((2x \mod 1,\beta y \mod 1)\). He shows that if \(\beta\) is a Parry number of PV-type, then \(K\) is a sofic \(T\)-invariant subset of the \(2\)-torus. Moreover, he obtains the result that if \(\beta\) is equal to the golden mean, then there exists a unique measure supported on \(K\) which is ergodic w.r.t. \(T\) and which has full Hausdorff dimension.