Conjugacy rigidity, cohomological triviality and barycentres of invariant measures
This thesis consists of three chapters and four appendices. Each chapter is a self-contained study of one topic in dynamical systems. Each chapter has its own notation. In chapter 1 we study conjugacies h between piecewise smooth conformal expanding Markov maps T1, T2. We prove that if h is piecewise continuous, satisfies an absolute continuity condition, and has essentially bounded partial derivatives, then it is as smooth as T1, T2. We use this result to give a new proof of part of Mostow's theorem on the rigidity of manifolds of constant negative curvature. In chapter 2 we study the cohomology of certain two-dimensional subshifts X with a semi-safe symbol. The simplest examples of such subshifts are the full shift and the golden mean shift. We prove the triviality of all locally constant co cycles on X taking values in a locally (residually finite) group. For real-valued cocycles we extend this result to the Holder category. In chapter 3 we consider the doubling map of the circle, and study the convex set n of barycentres of invariant measures. We prove that each interior point of n is the barycentre of an equilibrium state of a particular kind, and that this equilibrium state maximises entropy over all measures with this barycentre. We prove that any measure whose barycentre lies on the boundary an is not fully supported, and conjecture that its support has zero Hausdorff dimension. We conjecture further that an is non-differentiable at a countable dense set of points, the worst possible regularity for the boundary of a planar convex figure. Appendix A contains the proof of a technical lemma stated in section 1·3. Appendix B contains an alternative proof (due to de la Llave, Marco & Moriyon) of a theorem stated and proved in section 1·5. Appendix C contains numerical data to support the conjectures of chapter 3. Appendix D is a graphical plot of the data contained in Appendix C.