A note on forward iteration of inner functions

2022/06/01 by Ferreira, Gustavo Rodrigues · 3 citations
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.00374

Abstract

A well-known problem in holomorphic dynamics is to obtain Denjoy--Wolff-type results for compositions of self-maps of the unit disc. Here, we tackle the particular case of inner functions: if f_n:\mathbbD→\mathbbD are inner functions fixing the origin, we show that a limit function of f_n∘⋯∘ f₁ is either constant or an inner function. For the special case of Blaschke products, we prove a similar result and show, furthermore, that imposing certain conditions on the speed of convergence guarantees L¹ convergence of the boundary extensions. We give a counterexample showing that, without these extra conditions, the boundary extensions may diverge at all points of ∂\mathbbD.

Cited by

• Teichmüller spaces and normal forms associated to wandering domains
• The shrinking target and recurrence problem for non-autonomous systems
• A note on the boundary dynamics of holomorphic iterated function systems

Related

• Shrinking targets and recurrent behaviour for forward compositions of inner functions
• On the convergence of boundary points for hyperbolic inner functions
• Cyclic inner functions in growth classes and applications to approximation problems
• Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk
• Inner Functions, Composition Operators, Symbolic Dynamics and Thermodynamic Formalism