Dynamics of Smooth Hyperbolic Systems with Singularities

Abstract: This thesis is about dynamics of piecewise hyperbolic maps on bounded sets in the plane and on the interval. It is based on the following papers: T. Persson, A piecewise hyperbolic map with absolutely continuous invariant measure, Dynamical Systems: An International Journal, 21:3 (2006), 363--378 T. Persson, Absolutely continuous invariant measures for some piecewise hyperbolic affine maps, Preprints in Mathematical Sciences 2005:31, LUTFMA-5066-2005, ISSN 1403-9338 T. Persson, J. Schmeling, Dyadic Diophantine Approximation and Katok's Horseshoe Approximation, Preprints in Mathematical Sciences 2006:3, LUTFMA-5066-2006, ISSN 1403-9338 The first two papers are about two different classes of piecewise affine hyperbolic maps on bounded subsets of the plane. For both classes the tangent space can be decomposed into one expanding and one contracting subspace and the directions of those subspaces are the same for any point in the manifold. It is shown that, in the sense of parameters, for almost all maps in the two classes there exists an invariant measure that is absolutely continuous with respect to Lebesgue measure, provided that the map is sufficiently area expanding. It is also shown that these systems have exponential decay of correlations for Hölder continuous functions. There are also results on the topological entropy of maps for both classes of maps. The last paper is about dyadic Diophantine approximation --- the approximation of real numbers by rational numbers of which the denominators are a power of two. We calculate the dimensions of sets with different approximation speeds in the approximation. The paper also contains related results on the approximation of beta-shifts by finite type beta-shifts and the dimension of sets with different speeds of approximation. This is related to the dyadic Diophantine approximation since the the approximation of beta-shifts by finite type beta-shifts can be seen as an approximation of infinite beta-expansion of 1 by finite beta-expansions of 1. The thesis also contains results on a class of piecewise expanding map on the interval that can be obtained by a projection of maps from the first two articles. These maps are similar to the beta-expansion. The subshift associated to these maps are classified in terms of the orbit of the critical point and some connections between number theoretical properties of the expansion rate and the structure of periodic points are shown.