2020/10/07 by Guilhem Brunet, Brunet, Guilhem · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2010.03230
openalex publication_date 2020/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let m1 ≥ m2 ≥ 2 be integers. We consider subsets of the product symbolic sequence space (\0,⋯,m1-1\ × \0,⋯,m2-1\)ℕ^* that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres and Solomyak and using a fixed integer q ≥ 2. We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpinski carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely j = \lfloor logq ( (log(m1))/(log(m2)) ) \rfloor. We then generalize our results to the same subsets defined in dimension d ≥ 2. There, the situation is even more delicate and our formulas involve a collection of 2d-3 parameters.