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We introduce simple conditions ensuring that invariant distributions of aFeller Markov chain on a compact Riemannian manifold are absolutely continuouswith a lower semi-continuous, continuous or smooth density with respect to theRiemannian measure. This is applied to Markov chains obtained by randomcomposition of maps and to piecewise deterministic Markov processes obtained byrandom switching between flows.

On invariant distributions of Feller Markov chains with applications to dynamical systems with random switching