Some Topics On Piecewise Isometric Systems In Euclidean Spaces

Piecewise isometries (or PWIs for short) rooted from some electronic and elec-trical engineering questions which some scholars studied in eighties and nineties oflast century. Recently, some mathematicians investigated this area from a puremathematical point of view, and regarded a PWI as the natural generalization of aninterval exchange transformation (or IET for short) in order to found a theoreticalframework. In this thesis, the author discusses some basic questions of PWIs includ-ing the existence of periodic points and irrational points, the tangent-free propertyof periodic cell packings, dynamical complexity and so on.In Chapter one and two, the author introduces some main questions and pro-gresses in PWIs, and presents some basic notations of PWIs, such as piecewiseisometric map, coding, cell, global attraction and repelling and so on.Chapter three investigates the structure of periodic cells of higher dimensionalPWIs, and obtains that a periodic cell is symmetrical with respect with a center ora subspace, this is the generalization of the corresponding results of planar PWIs.Moreover, the author discusses the existence of periodic codings in planar PWIs, andgives a necessary and su?cient condition and a su?cient condition for the existenceof periodic codings. Some questions about irrational set are discussed in Chapterfour. It is shown that an irrational piecewise rotation must have an admissibleirrational coding and there exists a one-to-one relation between irrational pointsand admissible irrational codings.In Chapter five, the author discusses the tangent-free property of periodic cellpackings induced by general high-dimensional PWIs. The main results manifeststhat a periodic cell packing is tangent-free under some conditions. More importantly,the author applies the results to planar PWIs and obtains that the disk packinginduced by any piecewise irrational rotation is tangent-free, which strengthens allprevious results on this topic. At the same time, the author also studies the tangent-free property of a class of PWIs defined in product spaces.Chapter six discusses the dynamical complexity of PWIs. At first, the author investigates the essence of dynamical refinement and gives some helpful inequalitiesvia the Euler characteristic index formula. Then, the author discusses the complex-ity of planar PWIs according to this inequalities and gives a useful arithmetic toestimate the complexity of planar piecewise rational rotations. By this way, the au-thor calculates the complexity of Sigma-Delta map with rotation parameterθ= 35π.Chapter seven enumerates some questions deserving further investigation.