Holomorphic correspondences related to finitely generated rational semigroups

Abstract

In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on P 1 . Results on the iterative dynamics of such a correspondence can now be applied to the study of the rational semigroup. We focus on an invariant measure for the aforementioned correspondence -known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to analyse the distribution of repelling fixed points of a finitely generated rational semigroup, and to derive a sharp bound for the Hausdorff dimension of the Julia set of such a semigroup.

Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroups

“…Since µ(X \ Ω) = 0 and T (ψ n )(x) ≤ d t for all n ∈ Z + , by dominated convergence theorem, the limit of the left-hand side of (5.3) is ≥ 1 dt µ(F (U )) as n → ∞. Combining these observations, we obtain 1 dt µ(F (U )) ≤ µ(U ). Now, if A is a subset of X as in the hypothesis then, by regularity of the measure µ, we have…”

“…The idea of studying the dynamics of a finitely generated rational semigroup through the correspondence Γ G was introduced by Bharali-Sridharan in [1].…”

Dynamics of polynomial semigroups: measures, potentials, and external fields

“…Our main theorems, however, do not rely principally on Boyd's construction, nor do they rely on his methods. We shall not dwell on the reasons for this, but the interested reader is referred to [2,Remark 4.1] and to the fact that the semigroups that we shall consider are allowed to have degree-one elements. The semigroups we shall consider are described by the following Definition 1.3.…”

“…The next lemma is needed in the proof of Proposition 6.1. Recall, from the discussion in Section 1, that Proposition 6.1 establishes for the semigroups of our interest a result analogous to that in [2] -but which allows elements of G to have critical points in J(S). The following lemma is the key to dealing with the latter situation.…”

“…Since compositions of correspondences of the most general kind are not relevant to the proofs in this paper, we shall just make the following observations on the subject of composing two correspondences. (For readers who are more comfortable with complex analysis in one dimension, we refer to [2] for a more detailed discussion.) They are: (i ) The topological degree of a holomorphic correspondence Γ is the generic number of preimages of a point counted according to multiplicity.…”