Random holomorphic iterations and degenerate subdomains of the unit disk

Random Holomorphic Iterations and Degenerate Subdomains of the Unit Disk

Given a random sequence of holomorphic maps f1, f2, f3, . . . of the unit disk ∆ to a subdomain X, we consider the compositions Fn = f1 ◦ f2 ◦ . . . fn−1 ◦ fn. The sequence {Fn} is called the iterated function system coming from the sequence f1, f2, f3, . . . . We prove that a sufficient condition on the domain X for all limit functions of any {Fn} to be constant is also necessary. We prove the...

Random holomorphic iterations and degenerate subdomains

We first study necessary and sufficient conditions for subdomains of the unit disk to be degenerate under random holomorphic iterations; that is, any random composition of holomorpic maps from the domain to the subdomain has only constant functions as accumulation points. Sufficient conditions based on properties of the hyperbolic metric were known for a subdomain of the unit disk to be degener...

Ribbon R-trees and holomorphic dynamics on the unit disk

On Holomorphic Isometric Embeddings of the Unit Disk into Polydisks

We study the classification of holomorphic isometric embeddings of the unit disk into polydisks. As a corollary of our results, we can give a complete classification when the target is the 2-disk and the 3-disk. We also prove that the holomorphic isometric embeddings between polydisks are induced by those of the unit disk into polydisks.

On random points in the unit disk

Let n be a positive integer and λ > 0 a real number. Let Vn be a set of n points in the unit disk selected uniformly and independently at random. Define G(λ, n) to be the graph with vertex set Vn, in which two vertices are adjacent if and only if their Euclidean distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ lnn/n and let X be the number of isolated points in G(...