Profinite Structures and Dynamics

Surprising as it may be at first sight, there are a number of connections between the theories of finite semigroups and dynamical systems, both viewed in a broad sense. For instance in symbolic dynamics, ideas or analogies from the theory of finite automata find a natural setting for application in sofic systems [10, 24] and, even though not usually formulated in dynamical terms, the dynamical behavior of various operators on finite groups has been extensively studied. The purpose of this note is to review some further connections that have emerged recently driven mainly by work on finite semigroups and thus perhaps open the path to new investigations in this area. The main tool underlying our approach is found in profinite constructions, be it semigroups, groups, graphs or categories. Generally speaking, profinite structures are a way of encoding, with the help of an additional topological structure, common properties of a class of finite structures of the same type. This idea can be found in various areas, from Galois theory [17] to finite semigroup theory [6, 35, 4]. Results which are given without reference are announced here for the first time and will be proved elsewhere.