MULTIDIMENSIONAL STURMIAN SEQUENCES AND GENERALIZED SUBSTITUTIONS

Invertible substitutions and Sturmian sequences

It is known on the one hand that a Sturmian sequence can be generated geometrically by the intersections of a straight line with the unit grid in the plane, and on the other hand that fixed points of invertible substitutions are Sturmian. We give a new characterization of invertible substitutions, which allows to determine the straight line which generates the fixed point of a given invertible ...

Bidimensional Sturmian Sequences and Substitutions

Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms. The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitra...

Powers in Sturmian sequences

Substitutions on Multidimensional Sequen