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The M-set of λ exp(z)/z has infinite area

Part of: Holomorphic mappings and correspondences Complex dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  11 January 2016

Guoping Zhan  and
Liangwen Liao
Guoping Zhan
Affiliation:
Department of Mathematics, Zhejiang University of Technology, Hangzhou 310023, People's Republic of China, zhangp@zjut.edu.cn
Liangwen Liao
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China, maliao@nju.edu.cn
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Abstract

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It is known that the Fatou set of the map exp(z)/z defined on the punctured plane ℂ* is empty. We consider the M-set of λ exp(z)/z consisting of all parameters λ for which the Fatou set of λexp(z)/z is empty. We prove that the M-set of λexp(z)/z has infinite area. In particular, the Hausdorff dimension of the M-set is 2. We also discuss the area of complement of the M-set.

MSC classification

Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations 32H50: Iteration problems 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 217 , March 2015 , pp. 133 - 159
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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