Tohoku Mathematical Journal

Iteration of functions which are meromorphic outside a small set

Jian-Hua Zheng

Full-text: Open access

Abstract

In this paper, we investigate the dynamics of a broader class of functions which are meromorphic outside a compact totally disconnected set. We shall establish the connections between the Fatou components and the singularities of the inverse function and, accordingly, give sufficient conditions for the non-existence of wandering domains or Baker domains, and for the Julia set to be the Riemann sphere. Through the discussion of permutability of such functions, we shall construct several transcendental meromorphic functions which have Baker domains and wandering domains with special properties; for example, wandering and Baker domains with a critical value on the boundary and a wandering domain with the boundary being a Jordan curve (some such examples for entire functions were exhibited in other papers) and those of non-finite type which have no wandering domains.

Article information

Source
Tohoku Math. J. (2), Volume 57, Number 1 (2005), 23-43.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113234832

Digital Object Identifier
doi:10.2748/tmj/1113234832

Mathematical Reviews number (MathSciNet)
MR2113988

Zentralblatt MATH identifier
1158.37307

Subjects
Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]
Secondary: 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

Keywords
Meromorphic function iteration Fatou set Julia set

Citation

Zheng, Jian-Hua. Iteration of functions which are meromorphic outside a small set. Tohoku Math. J. (2) 57 (2005), no. 1, 23--43. doi:10.2748/tmj/1113234832. https://projecteuclid.org/euclid.tmj/1113234832


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