Bounded Fatou and Julia components of meromorphic functions
Copyright policy )Bounded Fatou and Julia components of meromorphic functions
AbstractWe completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering compacta using approximation theory.
- University of Liverpool United Kingdom
- Stony Brook University United States
- University of Salford United Kingdom
37B45, Fatou set, Mathematics - Complex Variables, Combinatorics and topology in relation with holomorphic dynamical systems, Julia set, Dynamical Systems (math.DS), Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable, Approximation in the complex plane, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets, Primary 37F10, wandering compacta, Fatou component, 54F15, Secondary 30D05, escaping set, FOS: Mathematics, meromorphic function, 37F10 (primary), 30D05, 37B45, 54F15 (secondary), Mathematics - Dynamical Systems, Complex Variables (math.CV), approximation theory, Meromorphic functions of one complex variable (general theory)
37B45, Fatou set, Mathematics - Complex Variables, Combinatorics and topology in relation with holomorphic dynamical systems, Julia set, Dynamical Systems (math.DS), Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable, Approximation in the complex plane, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets, Primary 37F10, wandering compacta, Fatou component, 54F15, Secondary 30D05, escaping set, FOS: Mathematics, meromorphic function, 37F10 (primary), 30D05, 37B45, 54F15 (secondary), Mathematics - Dynamical Systems, Complex Variables (math.CV), approximation theory, Meromorphic functions of one complex variable (general theory)
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