Piecewise smooth vector fields with sliding motion preserving measure

In this paper we establish conditions in order to obtain a set of trajectories of $n$-dimensional piecewise smooth vector fields that preserves measure even in the case where sliding motion is allowed. The key hypothesis is the occurrence of a sliding–escaping connection. As consequence, classical results from the ergodic theory of dynamical systems can be adapted for the context of piecewise smooth vector fields, just as the Poincaré’s Recurrence Theorem and the Birkhoff’s Theorem. Also, we apply the results for previous models of the literature.