Semigroup representations in holomorphic dynamics

Abstract

We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.

On hyperbolic cobordisms and Hurwitz classes of holomorphic coverings

“…The results in this section develop ideas in [3], and the Schreier representation of semigroups as treated in [2]. We start with a brief introduction to Schreier representations.…”

“…Finally, similar ideas allow us to consider the Hurwitz space as a representation space of a special class of holomorphic correspondences. This follows using results from [2] with [3]. So we can construct a Teichmüller space of correspondences of the form R −1 •R, called the deck correspondence associated to R. From [3] it follows that the Speisser class of R fibers over the Moduli space of the deck with fiber equivalent to the conformal Hurwitz class of R. Let G R = R −1 •R, C be the semigroup of holomorphic correspondences generated by the deck correspondence associated to R and constant maps.…”

“…So we can construct a Teichmüller space of correspondences of the form R −1 •R, called the deck correspondence associated to R. From [3] it follows that the Speisser class of R fibers over the Moduli space of the deck with fiber equivalent to the conformal Hurwitz class of R. Let G R = R −1 •R, C be the semigroup of holomorphic correspondences generated by the deck correspondence associated to R and constant maps. Consider the space X of all representations of G R into the semigroup of holomorphic correspondences on C. Then using results from [2], one can consider the Speisser class of a rational map R as a subspace of the connected component of X containing the identity representation.…”

“…By conjugation, f defines an automorphism of Rat(C) with composition as a product. By Proposition 8 in[2], the map f belongs to the group generated by P SL(2, C) and the absolute Galois group. Since ρ is continuous and orientation preserving we have f ∈ P SL(2, C).…”

Amenability and measure of maximal entropy for semigroups of rational maps

“…The dynamics of non-cyclic semigroups of rational maps iniciated by A. Hinkkanen and G. Martin in [11] is now an active area of research in holomorphic dynamics. Yet another approach is presented in [3] and [8].…”