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Sierpinski 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/1506892014Newton's method on Bring-Jerrard polynomials
https://ddd.uab.cat/record/150688
In this paper we study the topology of the hyperbolic component of the parameter plane for the Newton's method applied to n-th degree Bring-Jerrard polynomials given by P_n(z) = z^n-cz 1, \ c. For n=5, using the Tschirnhaus-Bring-Jerrard nonlinear transformations, this family controls, at least theoretically, the roots of all quintic polynomials. We also study a bifurcation cascade of the bifurcation locus by considering c . Campos, BeatrizFri, 06 May 2016 08:59:49 GMThttps://ddd.uab.cat/record/1506882014On Newton’s method applied to real polynomials
https://ddd.uab.cat/record/150565
It is known that if we apply Newton’s method to the complex function F(z) = P(z)e Q(z), with deg(Q) > 2, then the immediate basin of attraction of the roots of P has finite area. In this paper we show that under certain conditions on P, if deg(Q) = 1, then there is at least one immediate basin of attraction having infinite area. Cilingir, FigenFri, 06 May 2016 08:49:24 GMThttps://ddd.uab.cat/record/1505652012Brushing the hairs of transcendental entire functions
https://ddd.uab.cat/record/150538
Let f be a transcendental entire function of finite order in the EremenkoLyubich class B (or a finite composition of such maps), and suppose that f is hyperbolic and has a unique Fatou component. We show that the Julia set of f is a Cantor bouquet; i. e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if f ∈ B has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic with connected Fatou set, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function f ∈ B, a natural compactification of the dynamical plane by adding a “circle of addresses” at infinity. Baranski, KrzysztofFri, 06 May 2016 08:49:23 GMThttps://ddd.uab.cat/record/1505382012On the connectivity of the escaping set for complex exponential Misiurewicz parameters.
https://ddd.uab.cat/record/150462
Jarque i Ribera, XavierFri, 06 May 2016 08:30:51 GMThttps://ddd.uab.cat/record/1504622011A survey on the blow up technique
https://ddd.uab.cat/record/150433
The blow up technique is widely used in desingularization of degenerate singular points of planar vector fields. In this survey we give an overview of the different types of blow up and we illustrate them with many examples for better understanding. Moreover, we introduce a new generalization of the classical blow up. Álvarez Torres, María JesúsFri, 06 May 2016 08:30:49 GMThttps://ddd.uab.cat/record/1504332011On 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 Barcelona2011Non-landing hairs in Sierpinski curve Julia sets of transcendental entire maps
https://ddd.uab.cat/record/150407
We consider the family of transcendental entire maps given by fa (z) = a(z − (1 − a)) exp(z + a) where a is a complex parameter. Every map has a superattracting fixed point at z = −a and an asymptotic value at z = 0. For a > 1 the Julia set of fa is known to be homeomorphic to the Sierpi´ nski universal curve [19], thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing hairs with prescribed combinatorics embedded in the Julia set for all parameters a ≥ 3. We also study the relation between non-landing hairs and the immediate basin of attraction of z = −a. Even as each non-landing hair accumulates onto the boundary of the immediate basin at a single point, its closure, nonetheless, becomes an indecomposable subcontinuum of the Julia set. Garijo, AntoniFri, 06 May 2016 08:30:48 GMThttps://ddd.uab.cat/record/1504072011Wandering 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/1453122015Newton’s method on Bring-Jerrard polynomials
https://ddd.uab.cat/record/118295
Campos, BeatrizMon, 19 May 2014 13:50:21 GMThttps://ddd.uab.cat/record/1182952014Mètodes Matemàtics de l'Economia I
https://ddd.uab.cat/record/52866
Jarque i Ribera, XavierThu, 28 Jan 2010 18:40:50 GMThttps://ddd.uab.cat/record/528661993-94Joining polynomial and exponential combinatorics for some entire maps
https://ddd.uab.cat/record/52297
We consider families of entire transcendental maps given by Fλ,m (z) = λzm exp (z) where m ≥ 2. All these maps have a superattracting fixed point at z = 0 and a free critical point at z = −m. In parameter planes we focus on the capture zones, i. e. , we consider λ values for which the free critical point belongs to the basin of attraction of z = 0. We explain the connection between the dynamics near zero and the dynamics near infinity at the boundary of the immediate basin of attraction of the origin, thus, joining together exponential and polynomial behaviors in the same dynamical plane. Garijo, AntoniThu, 17 Dec 2009 19:52:50 GMThttps://ddd.uab.cat/record/522972010Models Matemàtics de l'Economia II
https://ddd.uab.cat/record/52171
Sans, NuriaMon, 14 Dec 2009 17:19:40 GMThttps://ddd.uab.cat/record/521711992-93Mètodes Matemàtics de l'Economia II
https://ddd.uab.cat/record/51852
Jarque i Ribera, XavierWed, 02 Dec 2009 17:01:47 GMThttps://ddd.uab.cat/record/518521993-94Models Matemàtics de l'Economia
https://ddd.uab.cat/record/51326
Jarque i Ribera, XavierWed, 18 Nov 2009 17:01:38 GMThttps://ddd.uab.cat/record/513261996-97Mètodes Matemàtics de l'Economia II
https://ddd.uab.cat/record/50134
Burgos, AlbertMon, 26 Oct 2009 18:33:40 GMThttps://ddd.uab.cat/record/501341994-95Models Matemàtics de l'Economia
https://ddd.uab.cat/record/48349
Jarque i Ribera, XavierTue, 06 Oct 2009 14:51:39 GMThttps://ddd.uab.cat/record/483491994-95Mètodes Matemàtics de l'Economia II
https://ddd.uab.cat/record/46917
Burgos, AlbertMon, 07 Sep 2009 15:44:58 GMThttps://ddd.uab.cat/record/469171994-95Mètodes Matemàtics de l'Economia I
https://ddd.uab.cat/record/46916
Dávila, JulioMon, 07 Sep 2009 15:43:47 GMThttps://ddd.uab.cat/record/469161994-95Models Matemàtics de l'Economia
https://ddd.uab.cat/record/45886
Jarque i Ribera, XavierThu, 16 Jul 2009 12:13:51 GMThttps://ddd.uab.cat/record/458861996-97Mediation incomplete information bargaining with filtered communication
https://ddd.uab.cat/record/45195
We analyze a continuous-time bilateral double auction in the presence of two-sided incomplete information and a smallest money unit. A distinguishing feature of our model is that intermediate concessions are not observable by the adversary: they are only communicated to a passive auctioneer. An alternative interpretation is that of mediated bargaining. We show that an equilibrium using only the extreme agreements always exists and display the necessary and sufficient condition for the existence of (perfect Bayesian) equilibra which yield intermediate agreements. For the symmetric case with uniform type distribution we numerically calculate the equilibria. We find that the equilibrium which does not use compromise agreements is the least efficient, however, the rest of the equilibria yield the lower social welfare the higher number of compromise agreements are used. Jarque i Ribera, XavierWed, 15 Jul 2009 08:38:55 GMThttps://ddd.uab.cat/record/451952006Chaotic dynamics in credit constrained emerging economies
https://ddd.uab.cat/record/45092
This paper analyzes the role of financial development as a source of endogenous instability in small open economies. By assuming that firms face credit constraints, our model displays a complex dynamic behavior for intermediate values of the parameter representing the level of financial development of the economy. The basic implication of our model is that economies experiencing a process of financial development are more unstable than both very underdeveloped and very developed economies. Our instability concept means that small shocks have a persistent effect on the long run behavior of the model and also that economies can exhibit cycles with a very high period or even chaotic dynamic patterns. Caballé, JordiWed, 15 Jul 2009 08:38:46 GMThttps://ddd.uab.cat/record/450922006