Random holomorphic iterations and degenerate subdomains of the unit disk

Abstract

Abstract. Given a random sequence of holomorphic maps f 1 , f 2 , f 3 , . . . of the unit disk ∆ to a subdomain X, we consider the compositionsThe sequence {F n } is called the iterated function system coming from the sequence f 1 , f 2 , f 3 , . . . . We prove that a sufficient condition on the domain X for all limit functions of any {F n } to be constant is also necessary. We prove that the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.

A Note on Random Holomorphic Iteration in Convex Domains

“…proved the following result. In [4] Keen and Lakic showed that the converse of (ii) also holds, as follows. In higher dimensions the story is different.…”

“…In [2] Beardon, Carne, Minda and Ng proved the following result: In [5] Keen and Lakic showed that also the converse of (2) holds:…”

“…In the case when D = D := {ζ ∈ C : |ζ| < 1}, degenerate subdomains have been completely characterized in terms of hyperbolic distance by Beardon et al . [2] and Keen and Lakic [4]. To state their results, we first introduce some terminology, which will also be needed later.…”

“…Given a holomorphic iterated function system, one is interested in knowing its asymptotical behavior, namely, to know the possible limits (in the compact-open topology for instance) of the sequence. In general such a question is rather difficult and one contents to know which conditions guarantee that every limit of {F j } is constant (we refer the reader to the papers [2] and [5] and bibliography therein).…”

Repeated compositions of Möbius transformations

“…The motivation for our study comes from discrete dynamical systems, where one is concerned with the iterates of a map f : X → X. In our more general context, given any collection F of self maps of X, we define a composition sequence generated by F to be a sequence (F n ), where F n = f 1 f 2 · · · f n and f i ∈ F. Composition sequences generated by sets of analytic self maps of complex domains have received much attention; see, for example, [2,6,13,14]. There has been particular focus on generating sets composed of Möbius transformations that map a disc within itself -see [1,4,5,12,15,16] -partly because of applications to the theory of continued fractions.…”