This special issue of Dynamical Systems: An International Journal is centred around an emerging area of dynamical systems that was the subject of a workshop in December 2005 at the University of Exeter, UK. The theme of this Special Issue is dynamical systems of geometric origin, more specifically those that are far from hyperbolicity, and their applications.
Geometric dynamics with singularities arise in a number of contexts, ranging from billiard systems, impact oscillators, models in digital signal processing and dynamics with discretized space. More abstract examples are provided by interval and polygon exchanges, and, more generally, by piecewise isometric systems. These systems display a form of dynamics that is intermediate between order and chaos: the key indicators of complexity grow algebraically rather than exponentially, and transport is, as a rule, anomalous (non-Gaussian).
Piecewise isometries give rise to partitions of phase space that have high complexity from minimal ingredients. These systems can readily be studied numerically (in significant cases by exploiting exact arithmetic with algebraic numbers), but their lack of hyperbolicity makes their analysis very difficult. The maps are often nonergodic, discontinuous and nonlinearly degenerate, and therefore many of the usual tools of smooth dynamics and ergodic theory are not directly applicable. Recent developments in these area have therefore relied on a disparate range of techniques such as geometric measure theory, topological dynamics, arithmetic number theory, algebraic dynamics and graph theory to mention a few. Several of these techniques are quite specialised and do not belong to mainstream dynamics. We hope that this Special Issue will stimulate others to consider their use where appropriate.
Arek Goetz
San Francisco State University, USA
Sebastian van Strien
University of Warwick, UK
Franco Vivaldi
Queen Mary, University of London, UK