Finitely constrained groups of maximal Hausdorff dimension

Abstract

We prove that if G_P is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group P of pattern size d, d>1, and if G_P has maximal Hausdorff dimension (equal to 1-1/2^{d-1}), then G_P is not topologically finitely generated. We describe precisely all essential pattern groups P that yield finitely constrained groups with maximal Haudorff dimension. For a given size d, d>1, there are exactly 2^{d-1} such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth d.

References in corpus (4)

• Classification of groups generated by 3-state automata over a 2-letter alphabet
• Hausdorff dimension of some groups acting on the binary tree
• Cellular automata between sofic tree shifts
• Profinite iterated monodromy groups arising from quadratic polynomials

Cited by in corpus (1)

• Restricted Hausdorff spectra of $q$-adic automorphisms