Dynamical Properties Of Planar Piecewise Isometries

Planar piecewise isometric systems have been investigated for many years in the?elds of physics, engineering and mathematics. In this thesis, we mainly apply dy-namical systems theory and numerical method to study the dynamical properties ofplanar piecewise isometric systems. We obtain some new results and provide manynew ideas and insights for further investigating planar piecewise isometric systems.We not only show theoretically some dynamical properties of planar piecewise iso-metric systems, but also combine the theory with practice and apply our theoreticalresults to some practical models. The investigation of planar piecewise isometricsystems has both theoretic signi?cance and practical interests.In details, the main contents of the dissertation are organized as follows: InChapter 1, we summarize the background and the recent progress in the researchof planar piecewise isometric systems. In Chapter 2, we introduce some elementaryspeculative knowledge. At ?rst, we introduce some basic notions including globalattractor, globally attracting map, exceptional set, topological entropy, dynamicalcomplexity, scattering, and so on. Secondly, we also show two models for planarpiecewise isometric systems, i.e., Goetz map and Sigma-Delta map. Finally, wediscuss some dynamical properties of piecewise isometric systems including globalattraction, the properties of invariant disk packing, invariant measures, and com-plexity, and so on. We discuss some problems about invariant measures for planarpiecewise isometric systems in Chapter 3. We show ?rstly some important invariantmeasures for planar piecewise isometric systems and then investigate some propertiesof them. In Chapter 4, we mainly discuss the periodicity and the dynamical com-plexity of planar piecewise isometric systems. We apply graph theory and symbolicdynamics method to study the dynamical complexity of planar piecewise isometricsystems. The relations between dynamical complexity and admissible ?nite codingsgenerated by the system will be discussed. At the same time, we calculate the dy-namical complexity of some planar piecewise isometric systems by applying our newresults. On the other hand, we also study the dynamics of the re?nement sequence of the partitions generated by a planar piecewise isometric system. We characterizethe scattering of the system through periodicity and complexity and use the con-cept of scattering to characterize higher order complexity of the planar piecewiseisometric system. Finally, we discuss the periodicity of ?nite codings generated bythe Sigma-Delta map and the relation between the periodicity and the complexityof this system. Chapter 5 summarizes the thesis and presents some problems forfurther research in the future.