Numeration systems and Markov partitions from self similar tilings

Abstract

Using self similar tilings we represent the elements of R n \mathbb {R}^n as digit expansions with digits in R n \mathbb {R}^n being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms.