Beyond Periodicity for Piecewise Contracting Maps

Details Beyond Periodicity for Piecewise Contracting Maps Beyond Periodicity for Piecewise Contracting Maps

2011-08-06

arxiv.org/abs/1108.1501v1

Mathematics

Dynamical Systems

Scientific paper

We study a wide class of dynamical systems defined by the iteration of piecewise contracting maps on compact and locally connected metric spaces. We consider non-isolated discontinuities and finitely many pieces of continuity in any finite or infinite dimensional space. We study the topological structure of the global attractors, non-wandering and limit sets associated to the orbits that do not intersect the frontiers of the different pieces of contraction. First, we obtain general results proving several new theorems and generalizing some old ones. We revisit the known strong hypothesis implying that the global attractor is made of a finite number of periodic orbits. We also give mild conditions on this class of maps that are sufficient for the total disconnection of the attractor and for its good recurrence properties. Second, we construct a series of examples for the mostly unknown case where the attractor intersects the frontiers. Thereby we show that the attractor may be finite, countably infinite or uncountable; it may include non-recurrent and also non-wandering points; and even when all the points in the attractor are non-wandering we prove that it does not necessarily coincide with the closure of all the admissible limit sets. This diversity of possible asymptotic behaviours is mostly interesting for models with milder structures than those commonly adopted.

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