Entropy and Gurevich Pressure for piecewise smooth vector fields

TL;DR

This paper connects classical discrete dynamical systems with non-smooth vector fields, calculating entropy and pressure, and analyzing the complexity and mixing properties of such systems.

Contribution

It introduces a method to compute Gurevich pressure and entropy for piecewise smooth vector fields using topological conjugacy and extends the analysis to non-smooth systems.

Findings

Calculated Gurevich pressure and entropy for non-smooth vector fields.

Established topological conjugacy with shift space for trajectories.

Related topological entropy at infinity to Hausdorff dimension of recurrent points.

Abstract

In this paper we provide a bridge between classical results concerning discrete dynamical systems and dynamical systems governed by nonsmooth vector fields. In fact, we obtain a set of piecewise smooth vector field trajectories where the time-one map is well defined. Afterwards, we obtain a topological conjugacy between the itinerary of a trajectory contained in this set and the one-sided shift space with a set of states countable with the discrete topology. We then use this fact to calculate the Gurevich pressure and entropy for non-smooth vector fields, and we also use the Lyapunnov function to estimate the mixing time. Furthermore, we relate the topological entropy at infinity of the global trajectories of these non-smooth vector fields to the Hausdorff dimension of the set of recurrent points that escape on average. Finally, we approach it from the point of view of cofinite topology, thus obtaining relevant examples.