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Tongues in Degree 4 Blaschke Products
https://ddd.uab.cat/record/169486
The goal of this paper is to investigate the family of Blasche products B_a(z)=z^3-a1- which is a rational family of perturbations of the doubling map. We focus on the tongue-like sets which appear in its parameter plane. We first study their basic topological properties and afterwords we investigate how bifurcations take place in a neighborhood of their tips. Finally we see how the period one tongue extends beyond its natural domain of definition. Canela Sánchez, JordiMon, 23 Jan 2017 15:21:48 GMThttps://ddd.uab.cat/record/1694862016On a family of degree 4 Blaschke products
https://ddd.uab.cat/record/169443
Canela Sánchez, JordiMon, 23 Jan 2017 15:21:45 GMThttps://ddd.uab.cat/record/169443Universitat de Barcelona,2015Hyperbolic entire functions with bounded Fatou components
https://ddd.uab.cat/record/169433
We show that an invariant Fatou component of a hyperbolic transcendental entire function is a Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our results are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values. Bergweiler, WalterMon, 23 Jan 2017 15:21:45 GMThttps://ddd.uab.cat/record/1694332015Sierpinski curve Julia sets for quadratic rational maps
https://ddd.uab.cat/record/150748
We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpi ́nski curve. Devaney, Robert L.Fri, 06 May 2016 08:59:53 GMThttps://ddd.uab.cat/record/1507482014On the connectivity of Julia sets of meromorphic functions
https://ddd.uab.cat/record/150738
We prove that every transcendental meromorphic map f with disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton’s method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions, whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton’s method for entire maps are simply connected, which solves a well-known open question. Baranski, KrzysztofFri, 06 May 2016 08:59:52 GMThttps://ddd.uab.cat/record/1507382014Absorbing sets and Baker domains for holomorphic maps
https://ddd.uab.cat/record/150689
We consider holomorphic maps f: U U for a hyperbolic domain U in the complex plane, such that the iterates of f converge to a boundary point of U. By a previous result of the authors, for such maps there exist nice absorbing domains W U. In this paper we show that W can be chosen to be simply connected, if f has parabolic~I type in the sense of the Baker--Pommerenke--Cowen classification of its lift by a universal covering (and is not an isolated boundary point of U). Moreover, we provide counterexamples for other types of the map f and give an exact characterization of parabolic~I type in terms of the dynamical behaviour of f. Baranski, KrzysztofFri, 06 May 2016 08:59:49 GMThttps://ddd.uab.cat/record/1506892014On the configuration of Herman rings of meromorphic functions
https://ddd.uab.cat/record/150555
We prove some results concerning the possible configurations of Herman rings for transcendental meromorphic functions. We show that one pole is enough to obtain cycles of Herman rings of arbitrary period and give a sufficient condition for a configuration to be realizable. Fagella Rabionet, NúriaFri, 06 May 2016 08:49:23 GMThttps://ddd.uab.cat/record/1505552012On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points II
https://ddd.uab.cat/record/150428
Following the attracting and preperiodic cases ([5]), in this paper we prove the existence of weakly repelling fixed points for transcendental meromorphic maps, provided that their Fatou set contains a multiply-connected parabolic basin. We use quasi-conformal surgery and virtually repelling fixed point techniques. Fagella Rabionet, NúriaFri, 06 May 2016 08:30:49 GMThttps://ddd.uab.cat/record/1504282011Connectivity of Julia sets of transcendental meromorphic functions
https://ddd.uab.cat/record/150409
Newton's method associated to a complex holomorphic function f is defined by the dynamical system Nf(z) = z – f(z) / f'(z). As a root-finding algorithm, a natural question is to understand the dynamics of Nf about its fixed points, as they correspond to the roots of the function f. In other words, we would like to understand the basins of attraction of Nf, i. e. , the sets of points that converge to a root of f under the iteration of Nf. Basins of attraction are actually just one type of stable component or component of the Fatou set, defined as the set of points for which the family of iterates is defined and normal locally. The Julia set or set of chaos is its complement (taken on the Riemann sphere). The study of the topology of these two sets is key in Holomorphic Dynamics. In 1990, Mitsuhiro Shishikura proved that, for any non-constant polynomial P, the Julia set of NP is connected. In fact, he obtained this result as a consequence of a much more general theorem for rational functions: If the Julia set of a rational function R is disconnected, then R has at least two weakly repelling fixed points. With the final goal of proving the transcendental version of this theorem, in this Thesis we see that: If a transcendental meromorphic function f has either a multiply-connected attractive basin, or a multiply-connected parabolic basin, or a multiply-connected Fatou component with simply-connected image, then f has at least one weakly repelling fixed point. Our proof for this result is mainly based in two techniques: quasiconformal surgery and the study of the existence of virtually repelling fixed points. We conclude the Thesis with an idea of the strategy for the proof of the case of Herman rings, as well as some ideas for the case of Baker domains, which is left as a subject for a future project. Taixés i Ventosa, JordiFri, 06 May 2016 08:30:48 GMThttps://ddd.uab.cat/record/150409Universitat de Barcelona2011A separation theorem for entire transcendental maps
https://ddd.uab.cat/record/145365
We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of ﬁnite order and with bounded set of singular values, or compositions thereof. Fix p ∈ N and assume that all dynamic rays which are invariant under f p land. An interior p-periodic point is a ﬁxed point of f p which is not the landing point of any periodic ray invariant under f p . Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above, we show that rays which are invariant under f p , together with their landing points, separate the plane into ﬁnitely many regions, each containing exactly one interior p−periodic point or one parabolic immediate basin invariant under f p . This result generalizes the Goldberg-Milnor Separation Theorem for polynomials [GM], and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel disks; that “hidden components” of a bounded Siegel disk have to be either wandering domains or preperiodic to the Siegel disk itself; or that there are only ﬁnitely many non-repelling cycles of any given period, regardless of the number of singular values. Benini, Anna MiriamTue, 12 Jan 2016 16:38:12 GMThttps://ddd.uab.cat/record/1453652015Wandering domains for composition of entire functions
https://ddd.uab.cat/record/145312
C. ~Bishop constructs an example of an entire function f in class B with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, f has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps f and g in class B such that the Fatou set of f g has a wandering domain, while all Fatou components of f or g are preperiodic. This complements a result of A. ~Singh and results of W. ~Bergweiler and A. Hinkkanen related to this problem. Fagella Rabionet, NúriaTue, 12 Jan 2016 16:38:08 GMThttps://ddd.uab.cat/record/1453122015On a Family of Rational Perturbations of the Doubling Map
https://ddd.uab.cat/record/145290
The goal of this paper is to investigate the parameter plane of a rational family of perturbations of the doubling map given by the Blaschke products. First we study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter a. We use techniques of quasiconformal surgery to explore the relation between certain members of the family and the degree 4 polynomials. In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type. Canela Sánchez, JordiTue, 12 Jan 2016 16:38:07 GMThttps://ddd.uab.cat/record/1452902015