Random iteration of analytic maps

Abstract

We consider analytic maps f j : D → D of a domain D into itself and ask when does the sequence f 1 •· · ·•f n converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence {w 1 , w 2 , . . . } in D whose values are not taken by any f j in D, and which is homogeneous in the sense that it comes within a fixed hyperbolic distance of any point of D. The situation for several complex variables is also discussed.

A Note on Random Holomorphic Iteration in Convex Domains

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

“…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).…”

Fourier frequencies in affine iterated function systems

“…While each finite algorithmic step can readily be pictured, not so for the iteration limit! And from the outset it may not even be clear whether or not a particular X is the attractor of an iterated function system (IFS); see, e.g., [3,4,10,15,24]. Moreover, far from all fractals fall in the affine IFS class.…”

Random holomorphic iterations and degenerate subdomains of the unit disk

“…A recent study of iterated function systems (see [3]) introduced new degenerate subdomains that are not relatively compact in ∆. It also considered more general iterated function systems formed from maps in Hol(Ω, X) where X ⊂ Ω are arbitrary plane domains such that Ω (and hence also X) admits ∆ as a universal cover.…”

“…Several recent articles are concerned with the study of iterated function systems and its applications, see for example, [3], [13], [14], [15]. We would like to thank Fred Gardiner for numerous helpful discussions and remarks on an earlier version of this paper and Jonathan Brezin for his editorial advice.…”