Qualitative theory of $p$-adic dynamical systems

We consider a class of dynamical systems over the $p$-adic number field: hierarchical dynamical systems. We prove a strong variant of the Poincaré theorem on the number of returns for such systems and show that hierarchical systems do not admit mixing. We describe hierarchical dynamical systems over the projective line and present an example of a nonhierarchical $p$-adic system that admits mixing: the $p$-adic baker's transformation.