Complex dynamics of the limit periodic system Fn(z) = Fn−1(⨍n(z)), ⨍n → ⨍

摘要

The sequence {Fn(z)} is one kind of generalization of limit periodic continued fractions. The convergence/divergence of {Fn(z)} relative to that of the iterative sequence {Fn(z)} is determined when ⨍n→⨍ uniformly on a domain S. When ⨍ maps the closure of S into S,Fn(z)→λ for all zϵS. If ⨍n → ⨍ rapidly on S, the sequence {Fn} uniformly shadows{⨍n}(|Fn(z)−⨍n(z)|<ε∀n,∀zϵS), and the two sequences show similar complex dynamics. A priori uniform “tail end” bounds are provided: |⨍n + 1 ō ⋯ō ⨍n + m(z)−gn+1 ō ⋯ō gn + m (z)| < εn ∀m, where, for example, the first expansion may represent a modified continued fraction and the second another limit periodic structure with gn → ⨍.