Dynamics of Piecewise Isometries

We begin a systematic study of Euclidean Piecewise Isometric Dynamical Systems (p.i.d.s.) with a particular focus on the interplay between geometry, and symbolic dynamics , and the group of isometries associated with p.i.d.s. We investigate various aspects of the dynamical information contained in the coding: symbolic growth and the periodic behavior of codings and cells. This theoretical investigation is motivated by the many examples of piecewise isometric dynamical systems found recently in the literature. Piecewise isometric dynamical systems are direct generalizations of interval exchange transformations to non-invertible, higher dimensional maps. 1. Introduction In this manuscript we begin underlying basic foundations of the theory of Euclidean Piecewise Isometric Dynamical Systems. The aim of this theory is to study the long term behavior of states in a dynamical system for which the local generating maps are isometries. Our motivation for this article includes the many intriguing examples of piecewise isometries with their mysterious behavior and beautiful computer graphics. Systems of piecewise isometries have been recently linked to the dynamics of electronic Moreover, these systems generalize well known and studied interval exchanges to a class of Euclidean two-dimensional piecewise isometries. Piecewise isometries appear in a variety of contexts and have been recently extensively studied as The structure of the article is as follows. In the rst ve sections, we study the symbolic growth of symbolic sequences associated to p.i.d.s. The main result is in section 4 and it relates the growth of the associated semigroup of isometries and the growth of symbolic words.