Some Counterexamples in Dynamics of Rational Semigroups

Decidability and Tameness in the Theory of Finite Semigroups

2 1 Ju l 2 00 9 Random complex dynamics and semigroups of holomorphic maps ∗

We investigate the random dynamics of rational maps on the Riemann sphere Ĉ and the dynamics of semigroups of rational maps on Ĉ. We see that the both fields are related to each other very deeply. We investigate spectral properties of transition operators and the dynamics of associated semigroups of rational maps. We define several kinds of Julia sets of the associated Markov processes and we s...

4 D ec 2 00 8 Random complex dynamics and semigroups of holomorphic maps ∗

We investigate the random dynamics of rational maps on the Riemann sphere Ĉ and the dynamics of semigroups of rational maps on Ĉ. We see that the both fields are related to each other very deeply. We investigate spectral properties of transition operators and the dynamics of associated semigroups of rational maps. We define several kinds of Julia sets of the associated Markov processes and we s...

The Dynamics of Semigroups of Rational Functions I

This paper is concerned with a generalisation of the classical theory of the dynamics associated to the iteration of a rational mapping of the Riemann sphere, to the more general setting of the dynamics associated to an arbitrary semigroup of rational mappings. We are partly motivated by results of Gehring and Martin which show that certain parameter spaces for KJeinian groups are essentially t...

The Equilibrium States for Semigroups of Rational Maps

In this paper, we frequently use the notation from [S1]. A “rational semigroup” G is a semigroup generated by non-constant rational maps g : CI → CI, where CI denotes the Riemann sphere, with the semigroup operation being functional composition. For a rational semigroup G, we set F (G) := {z ∈ CI | G is normal in a neighborhood of z} and J(G) := CI \ F (G). F (G) is called the Fatou set of G an...