Compositions of contractions

Boundary points as limit functions of iterated holomorphic function systems

“…Lorentzen [7] and Gill [3] showed that if X is relatively compact in ∆, then every iterated function system converges to a unique constant which is inside X not on its boundary. Therefore, non-relative compactness of X in ∆ is a necessary condition in order to have a boundary point of X as a limit function of an iterated function system.…”

Random iterations, fixed points and invariant CRF-horospheres in complex Banach spaces

“…The important questions concern the behaviour of the random iterations f 1 • f 2 • · · · • f n and f n • f n−1 • · · · • f 1 , where f n ∈ Hol(G, G) and n ∈ N [11,12,[18][19][20][21]31,35,44,47,53], the iterative behaviour of an individual map f [n] = f • · · · • f (n times) [1][2][3][5][6][7][8]13,14,[26][27][28][29][30]33,34,36,37,41,42,45,49] and the existence of fixed points for f ∈ Hol(G, G) [13,24,40,43,48] and the existence of invariant horocycles, ellipsoids or horospheres in G determined by Kobayashi metrics k G [1][2][3][4][5][6][7][8][15][16]…”

Random holomorphic iterations and degenerate subdomains of the unit disk

“…One result of Lorentzen and Gill is Theorem GL ([9], [12]). If an iterated function system is formed from functions in Hol(∆, X) where X is relatively compact in ∆, then the system F n converges locally uniformly in ∆ to a unique constant.…”