Symbolic dynamics and self-similar groups

Self-similar groups and permutational bimodules are used to study combinatorics and symbolic dynamics of expanding self-coverings. We describe functors between the category of contracting self-similar groups and the category of expanding self-coverings (with appropriate morphisms). These functors transform some questions in dynamical systems to questions in algebra. As examples we show how some plane filling curves (in particular the original Peano curve) can be interpreted in terms of embeddings of self-similar groups.