Abstract
The starting point of the research developed in this thesis was the work done by Arek Goetz in his PhD thesis [Dynamics of Piecewise Isometrics, University of Illinois, 1996). Following his dissertation, we have considered the simple example of a piecewise rotation in two convex atoms defined in the whole plane (now commonly known as the Goetz map) as our main source of motivation. The first main achievement of our work was the construction of a new family of polygonal sets which are, in fact, global attractors. These examples are very similar in nature to the Sierpinski-gasket triangle presented in Goetz [1998c]. The next natural step was to argue that the definition of a piecewise isometric attractor is not entirely suitable in this context due to the lack of uniqueness. Under a new definition of attractor, some results are proved involving the properties of quasi-invariance and regularity existing in all examples available in the literature on the Goetz map. Following that, we attempt to generalise the results of Goetz regarding periodic cells and periodic points to unbounded spaces. We prove that there is a fundamental discrepancy between piecewise rotations in odd and even dimensions. In the odd-dimensional case the existence of periodic points is rare; hence those that exist must be unstable under almost all perturbations, whereas in even dimensions periodic points are stable for a prevalent set of piecewise rotations. Furthermore, if a piecewise rotation is such that the free monoid generated by the linear parts of the induced rotations does not contain the identity map then it follows trivially that all diverging orbits must be irrationally coded. This implies, together with a result on the coding of the open connected components in the complement of the closure of the exceptional set, that there exist examples of piecewise isometries possessing irrational cells with positive measure. This scenario was not considered previously in the results proved by Goetz. Using a common recurrence argument and Goetz's characterisation of the closure of the exceptional set (see for instance, Goetz [2001]) we prove that every recurrent point [i.e., such that w(x) ? ?) must be rationally coded. Given an invertible piecewise isometry in a compact space we also show that the closure of the exceptional set equals that of its inverse. This sustains the common idea that forward and backward iterates of the discontinuity yield similar graphics. In the context of the Goetz map we have also investigated the appearance of symmetric patterns when plotting the closure of the exceptional set. Although the Goetz map is by its nature discontinuous, the existence of symmetry is still possible under the broader framework of almost-everywhere-symmetry. Finally, we briefly note that the local symmetry properties of symmetric patterns arising in the invertible Goetz map are in part due to the existence of a reversing-symmetry, which generates piecewise continuous reversing symmetries under iteration of the original Goetz map.