Some Aspects of Thermodynamic Formalism of Piecewise Smooth Vector Fields

Abstract

In this paper we study some aspects of thermodynamic formalism, more specifically topological pressure and, as a consequence, topological entropy for piecewise smooth vector fields, using topological conjugation with shift maps and the Perron-Frobenius Operator. Some relationships between entropy and Hausdorff dimensions are also investigated. As consequences of our results, we obtain planar piecewise smooth vector fields with topological entropy and Hausdorff dimension equal to logα$$\log \alpha $$ for all α∈(1,2]$$\alpha \in (1, 2]$$. We stress that, for the best of our knowledge, it is the first time in the literature where the topological pressure and the Hausdorff dimension of a piecewise smooth vector field is obtained. Moreover, the obtainment of topological entropy given equal to a real number logα$$\log \alpha $$, for all α∈(1,2]$$\alpha \in (1, 2]$$, is an extension of previous results where the topological entropy of the planar piecewise smooth vector fields considered are logk$$\log k$$, for k a positive integer.