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• W2534729756 abstract "A hyperbolic transcendental entire function with connected Fatou set is said to be disjoint It is known that a disjoint-type function provides a model for the dynamics near infinity of all maps in the same parameter space; hence a good understanding of these functions has implications in wider generality. Our goal is to study the topological properties of the sets of entire functions of disjoint type. In particular, we give a detailed description of the topology of their connected components. More precisely, consider a Julia C of such a function, i.e. the closure in the Riemann sphere of a component of the set. We show that infinity is a terminal point of C, and that C has span zero in the sense of Lelek; under a mild geometric assumption on the function C is arc-like. (Whether every span zero continuum is also arc-like was a famous question in continuum theory, only recently resolved in the negative.) Conversely, we construct a single disjoint-type entire function with the remarkable property that each arc-like continuum with at least one terminal point is realised as a continuum. The class of arc-like continua with terminal points is uncountable. It includes, in particular, the sin(1/x)-curve, the Knaster buckethandle and the pseudo-arc, so these can all occur as continua of a disjoint-type entire function. We give similar descriptions of the possible topology of continua that contain periodic points or points with bounded orbits, and answer a question of Baranski and Karpinska by showing that continua need not contain points that are accessible from the Fatou set. Furthermore, we construct an entire function whose set has connected components on which the iterates tend to infinity pointwise, but not uniformly. This is related to a famous conjecture of Eremenko concerning escaping sets of entire functions." @default.
• W2534729756 created "2016-10-28" @default.
• W2534729756 creator A5035883645 @default.
• W2534729756 date "2016-10-20" @default.
• W2534729756 modified "2023-09-27" @default.
• W2534729756 title "Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture" @default.
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