Semigroup representations in holomorphic dynamics
Carlos Cabrera Peter Makienko Peter Plaumann
Discrete & Continuous Dynamical Systems - A 2013, 33(4): 1333-1349 doi: 10.3934/dcds.2013.33.1333
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.
keywords: Semigroup representations complex polynomials holomorphic dynamics holomorphic correspondences. rational maps
Ruelle operator and transcendental entire maps
Patricia Domínguez Peter Makienko Guillermo Sienra
Discrete & Continuous Dynamical Systems - A 2005, 12(4): 773-789 doi: 10.3934/dcds.2005.12.773
We calculate the Ruelle operator of a transcendental entire function $f$ having only a finite set of algebraic singularities. Moreover, under certain topological conditions on the postcritical set we prove (i) if $f$ has a summable critical point, then $f$ is not structurally stable and (ii) if all critical points of $f$ belonging to Julia set are summable, then there do not exist invariant lines fields on the Julia set.
keywords: entire functions Ruelle operator Fatou set Julia set invariant line fields.

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