This thesis consists of two independent chapters. Both present results in the field of dynamical systems. In the first chapter we study abstract adding machines and their occurrence in unimodal maps of the interval. For unimodal maps with no aperiodic homtervals we characterize completely when adding machines occur. We also discuss their importance in relation to the boundary of positive topological entropy for two-parameter families of diffeomorphisms of the disc. In the second chapter we prove existence of a distinguished set of geodesics on orientable Riemannian surfaces with geodesic boundary and negative Euler characteristic. This result allows us to construct a semi-equivalence between a subset of the unit tangent bundle of the surface with the arbitrary Riemannian metric and the unit tangent bundle given by the standard hyperbolic metric on this surface. The result is analogous to one of Morse [Ml] for surfaces without boundary. We give a new proof of Morse’s result using a method similar to the proof of our new result.