Campos de vetores suaves por partes: preservação de medida, pressão topológica e dinâmica simbólica

Abstract

The study of piecewise smooth vector fields (PSVFs) has been consolidated in recent years not only because of the beauty of the theoretical results, but also because of the proximity of this area to applied sciences such as mechanics, engineering, electronics and biology, in addition to social sciences and economical. The main difference between PSVFs and smooth vector fields is the fact that there may not be unique the trajectory passing through each point a PSVF. With the existence of chaos, we can look for ways to calculate the topological entropy, since entropy estimates how chaotic the environment is system. In this work we follow this line of investigation and obtain a set of piecewise smooth vector field trajectories where the application of time one is well defined. In this way, we obtain a conjugacy between the itinerary of a trajectory contained in this set and sequences over a finite set of symbols. Thus, we study some aspects of thermodynamic formalism, more specifically topological pressure and, consequently, topological entropy for piecewise smooth vector fields, using topological conjugacy with one-sided shifts and the Ruelle-Perron-Frobenius Operator. Some relations between entropy, Hausdorff dimension and Minkowski dimension are also presented. In this sense, when the pressure is zero, we can use the Markov chain theory together with the Ruelle-Perron-Frobenius operator to calculate the relaxation time and estimate the mixing time for PSVFs. Finally, we introduce the concept of sliding-escaping connection for piecewise smooth vector fields and establish conditions in order to obtain a set of trajectories that preserves measure even in the case where sliding motion is allowed. As consequence, classical results from the ergodic theory of dynamical systems can be adapted for the context of piecewise smooth vector fields with a sliding-escaping connection, namely, the Poincare’s Recurrence Theorem and the Birkhoff’s Ergodic's Theorem.