*(English)*Zbl 0964.37009

The subject of the paper is Euclidean piecewise isometric dynamical systems (p.i.d.s.). A piecewise isometry is a pair $(T,\mathcal{P})$, where $T:X\to X$ is a map such that its restriction to ${P}_{i}$, $i=0,\cdots ,r-1$ is a Euclidean isometry. Here $X$ is a subset of ${\mathbb{R}}^{N}$ and $\mathcal{P}=\{{P}_{0},\cdots ,{P}_{r-1}\}$ $(r>1)$ is a finite partition of $X$. Such systems are direct generalization of interval exchange transformations to non-invertible, higher-dimensional maps.

Considering geometrical properties of p.i.d.s. and symbolic dynamics the author describes the relation between the growth of the associated semigroup of isometries and the growth of symbolic words. The notion of entropy for piecewise isometry is introduced and expressed in terms of the growth of symbolic words. Furthermore, the author studies necessary conditions for a p.i.d.s. to generate all possible finite words. Finally, some remarks on the interplay between symbolic codings, periodic points, and the geometry of cells (i.e. sets following the same coding pattern) are given. A number of results included are generalizations of results for interval exchangers.