Symbolic dynamics and self-similar groups

Abstract

Abstract. 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.

Mating, paper folding, and an endomorphism of PC^2

“…We present here, in a very condensed form, the main definitions and results of the theory of self-similar and iterated monodromy groups. For a more detailed account, see [13,15,17,19].…”

“…The correspondence between the contracting self-similar groups and expanding self-coverings is functorial in a precise way, see [15]. For example, any embedding of self-similar contracting groups (preserving self-similarity) induces a semi-conjugacy of their limit dynamical systems.…”

Exponential growth of some iterated monodromy groups

“…The hypothesis of non‐renormalizability rules out the counterexamples that arise from tuning , a reverse operation to renormalization . The tuning operation allows to construct examples of polynomials with dendrite Julia set whose s are of exponential growth, see [, Section 5.5]. However, those maps will be renormalizable.…”