Self-similarity in piecewise isometric systems

Abstract

This paper deals with piecewise isometries and their scaling properties. We aim to present a general framework in which piecewise isometric dynamical systems can be fully or partially renormalized. We precisely describe the dynamics on fractal invariant sets by means of a conjugacy with substitution dynamical systems and Bratteli diagrams. We describe their geometrical properties in terms of graph-directed constructions and we compute their Hausdorff dimension. We also present an example of application to a continuous family of piecewise rotations.