2008/11/22 by Hiroki Sumi, Sumi, Hiroki
Mathematics · #30D05 #37F10 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Probability (math.PR) #math.CV #math.DS #math.GT #math.PR #msc:30D05 #msc:37F10
paper · pdf · doi:10.48550/arxiv.0811.3664
Published in Discrete and Continuous Dynamical Systems - Series A, Vol. 29, No. 3, 2011, 1205--1244. 39 pages, 2 figures. Some typos are fixed. See also http://www.math.sci.osaka-u.ac.jp/~sumi/
openalex publication_date 2008/11/22 · arxiv created 2011/01/19 · arxiv updated 2011/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if A and B are two connected components of the Julia set, then one of A and B surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any n∈ ℕ ∪ \ℵ0\ , there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to n. Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated.