Spectral theory and geometric representation of substitutions

Abstract

From geometry to symbolic dynamics. As explained in Chap. 5, symbolic dynamical systems were first introduced to better understand the dynamics of geometric maps. Indeed, by coding the orbits of a dynamical system with respect to a cleverly chosen finite partition indexed by the alphabet A, one can replace the initial dynamical system, which may be difficult to understand, by a simpler dynamical system, that is, the shift map on a subset of A^ℕ.

This old idea was used intensively, up to these days, particularly to study dynamical systems for which past and future are disjoint, such as toral automorphisms or pseudo-Anosov diffeomorphisms of surfaces. These systems with no memory, whose entropy is strictly positive, are coded by subshifts of finite type, defined by a finite number of forbidden words. Some very important literature has been devoted to their many properties (see [265]). The partitions which provide a good description for a topological dynamical system, leading to a subshift of finite type, are called Markov partitions (a precise definition will be given in Sec. 7.1).

This chapter has been written by A. Siegel