Introduction to Discontinuous Dynamical Systems

The topics of this book are dynamical behaviour of a class of discontinuous maps including piecewise linear maps on the 2-torus and planar piecewise isometries, some preliminary symbolic dynamics approach to these maps, and some aspects of symbolic dynamics itself. At first, a class of piecewise linear parabolic maps on the torus are discussed. Then some area-preserving piecewise linear discontinuous maps of the plane are examined. Symbolic descriptions of the dynamics and admissibility conditions for itineraries of a class of linear maps on the 2-torus and some planar piecewise isometries are discussed. For the symbolic dynamical system itself, scrambled sets of shift maps are discussed, and some problems related to maximal scrambled sets are also discussed. For the subshifts generated by the orbit closures of certain non-periodic recurrent points, their chaotic properties are investigated. Moreover, a general and systematic discussion of various symbolic representations of iterated maps through subshifts are presented, and examples of partitions and representations of some discontinuous maps are given.