Abstract

The present work is divided into two parts. The first part is dedicated to piecewise contracting maps defined over a compact interval. In this context, we prove the existence of a decomposition of the asymptotic dynamics into minimal components, which are either periodic orbits or Cantor sets. Furthermore, we use said decomposition to prove that these piecewise contracting maps admit ergodic measures. In the second part, we explore and adapt some classical tools for the study of a certain class of piecewise continuous dynamical systems of the interval. We begin by defining an appropriate metrizable topology over the space of piecewise Lipschitz maps. We prove that said metric space satisfies interesting properties such as enlarge the concept of closeness given by the uniform metric and being a Baire space. Finally, we define a new concept of topological entropy adapted to continuous piecewise maps, proving that such a new concept is preserved by conjugacies.