  We study a class of Piecewise Deterministic Markov Processes with state space
Rd x E where E is a finite set. The continuous component evolves according to a
smooth vector field that is switched at the jump times of the discrete
coordinate. The jump rates may depend on the whole position of the process.
Working under the general assumption that the process stays in a compact set,
we detail a possible construction of the process and characterize its support,
in terms of the solutions set of a differential inclusion. We establish results
on the long time behaviour of the process, in relation to a certain set of
accessible points, which is shown to be strongly linked to the support of
invariant measures. Under H\"ormander-type bracket conditions, we prove that
there exists a unique invariant measure and that the processes converges to
equilibrium in total variation. Finally we give examples where the bracket
condition does not hold, and where there may be one or many invariant measures,
depending on the jump rates between the flows.
