Category: Working Seminar on Complex Dynamics

Date: 24.07.20

Time: 11:30 - 12:30

Additional Information:

Local connectivity is an important concept in holomorphic dynamics. The local

connectivity of a Julia set is a sign that, despite being fractal, the Julia set is understandable.

If the boundary of a simply connected Fatou component is locally connected, it even means that

this boundary can be parametrized by the unit circle. One of the flag conjectures in holomorphic dynamics

is in fact the local connectivity of the Mandelbrot set.

In this talk we explain what is known about local connectivity of Julia sets in different contexts and

we show in detail the proof of local connectivity of the simplest case. We also emphasize the difficulties

in other cases and state some new results for transcendental functions.

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Category: Working Seminar on Complex Dynamics

Date: 02.07.20

Time: 16:00 - 16:30

Additional Information:

The main goal of this talk is to understand and prove, using recently developed techniques,

Shishikura’s result on the connectivity of the Julia set of the Newton map of polynomials.

To do so, we first present a set of preliminary tools that contain normal families, conformal

representations and proper maps, among others. It is followed by a study of rational complex

dynamical systems, some results on the existence of fixed points of meromorphic maps and

it is concluded by what is the cornerstone of this project: the proof of the connectivity of the

Julia set of Newton maps of polynomials.

Category: Working Seminar on Complex Dynamics

Date: 02.07.20

Time: 16:30 - 17:00

Additional Information:

L'objectiu principal de la xerrada és demostrar el Teorema de Denjoy-Wolff, que tracta la iteració de funcions holomorfes al disc unitat. El teorema afirma que donada una funció holomorfa al disc unitat o bé és conjugada a una rotació, o bé hi ha un únic punt cap al qual tendeixen totes les òrbites. També demostrarem que, en un entorn d'aquest punt, la funció és conjugada a una transformació conforme. Finalment, utilitzarem aquests resultats per a classificar les components de Fatou periòdiques de funcions enteres i per a classificar els dominis de Baker en funció de la seva dinàmica.

]]>Category: Working Seminar on Complex Dynamics

Date: 25.06.20

Time: 16:00 - 17:00

Additional Information:

Let U be a Fatou component of a transcendental entire function. If U is not eventually periodic then it is called a wandering domain. Although Sullivan's celebrated result showed that rational maps have no wandering domains, transcendental entire functions can have wandering domains. The first wandering domain of oscillating type was constructed by Eremenko and Lyubich in 1987. Motivated by their construction and the recent classification of simply connected wandering domains obtained by Benini, E., Fagella, Rippon and Stallard, we give a general technique, based on Approximation Theory, for the construction of bounded oscillating wandering domains. We show that this technique can be used to produce examples of oscillating wandering domains of all six different types that arise by the classification. This is joint work with P. Rippon and G. Stallard.

]]>Category: Working Seminar on Complex Dynamics

Date: 18.06.20

Time: 16:00 - 17:00

Additional Information:

In this talk we present the state of the art of the Newton's method defined in $\mathbb R^n$ as a global dynamical system.

]]>Category: Working Seminar on Complex Dynamics

Date: 11.06.20

Time: 16:00 - 17:00

Additional Information:

Abstract:

In this talk we will present a family of rational maps obtained by applying the Chebyshev-Halley family of root-finding algorithms to $z^3-1$. We will show that, for certain parameters, these maps can be studied from the point of view of singular perturbations. We will also show that, for such parameters, the corresponding Julia sets can be locally connected and contain Cantor sets of quasicircles.

]]>Category: Working Seminar on Complex Dynamics

Date: 04.06.20

Time: 17:15 - 18:15

Additional Information:

A function $f$ is a transcendental self-map of the punctured plane if $f:\mathbb{C}^*\to\mathbb{C}^*

Category: Working Seminar on Complex Dynamics

Date: 26.05.20

Time: 15:30 - 16:30

Additional Information:

We will focus on the mutual relations between singular values,

periodic rays and periodic orbits in transcendental dynamics.

It is classically known that the existence of non-repelling periodic

points is connected to the presence of a singular value 'near by'- for

example, attracting and parabolic basins need to contain at least one

singular value. For repelling periodic points, singular values come

into play when the periodic point in question is not the landing point

of any periodic ray.

In this talk we will show that under our assumptions, it is possible

to associate a specific singular orbit to every non-repelling cycle,

as well as to every repelling cycle whose points are not landing

points of periodic rays.

This gives a version of the Fatou-Shishikura inequality which takes

into account such repelling cycles.

This is joint work with N. Fagella.

Category: Working Seminar on Complex Dynamics

Date: 12.05.20

Time: 10:00 - 11:00

Additional Information:

In this talk we will describe how Fatou coordinates may be used to study of the real dynamics of a given biological problem. More specifically, we will analyse how the multiplier of the two complex repelling fixed points that appear after a saddle-node bifurcation may affect the number of iterates that a real point spends to go through the former parabolic fixed point.

]]>Category: Working Seminar on Complex Dynamics

Date: 28.04.20

Time: 10:00 - 11:00

Additional Information:

In this talk we will present a family of rational maps obtained by applying the Chebyshev-Halley family of root-finding algorithms to $z^3-1$. We will show that, for certain parameters, these maps can be studied from the point of view of singular perturbations. We will also show that, for such parameters, the corresponding Julia sets can be locally connected and contain Cantor sets of quasicircles.

]]>Category: Working Seminar on Complex Dynamics

Date: 21.04.20

Time: 10:00 - 11:00

Additional Information:

We consider on the Riemann sphere the vector field defined by a rational map. We study the properties of the separatrix graph associated to this kind of vector fields.

]]>Category: Working Seminar on Complex Dynamics

Date: 14.04.20

Time: 10:00 - 11:00

Additional Information:

We consider on the Riemann sphere the vector field defined by a rational map. We study the properties of the separatrix graph associated to this kind of vector fields.

]]>Category: Working Seminar on Complex Dynamics

Date: 07.04.20

Time: 10:00 - 11:00

Additional Information:

We present some of the well known Koebe distortion Theorems and the main applications to holomorphic dynamics.

]]>Category: Working Seminar on Complex Dynamics

Date: 31.03.20

Time: 10:00 - 11:00

Additional Information:

We present some of the well known Koebe distortion Theorems and the main applications to holomorphic dynamics.

Category: Working Seminar on Complex Dynamics

Date: 10.03.20

Time: 12:00 - 13:00

Additional Information:

We study the theorem based on the article by Branner and Douady, that present the relation between the limb M_1/2 and the limb M_1/3 of the Mandelbrot set. The result is obtained by surgery.

]]>Category: Working Seminar on Complex Dynamics

Date: 02.03.20

Time: 17:00 - 18:00

Additional Information:

In the first talk it was described the choice of the unperturbed family that we have studied. We continue with the description of the dynamical plane for perturbed maps and the exact achievable connectivities of Fatou components.

]]>Category: Working Seminar on Complex Dynamics

Date: 24.02.20

Time: 17:10 - 18:00

Additional Information:

We discuss certain conditions for coincident Fatou, Julia and escaping sets of a holomorphic semigroup and its proper subsemigroups.

]]>Category: Working Seminar on Complex Dynamics

Date: 15.01.20

Time: 09:30 - 10:30

Additional Information:

The connectivity locus $M$ for 2-gon complex trees $T\{c,-c\}$ turns out to be the closure of all roots (contained in the unit disk) of polynomials with coefficients 1, -1, and 0, starting always with 1. If we select only roots of polynomials with coefficients equal to 1 or -1 then we obtain a subset $M_0$ which is tightly connected to the Thurston Set introduced in Bill Thurston’s last paper "Entropy in Dimension One". In this talk we will explore the internal structure of $M$ that follows naturally from the node-to-node connectivity theorem of 2-gon complex trees $T\{c,-c\}$.

]]>Category: Working Seminar on Complex Dynamics

Date: 08.01.20

Time: 12:30 - 13:30

Additional Information:

We study the theorem based on the article by Branner and Douady, that present the relation between the limb M_1/2 and the limb M_1/3 of the Mandelbrot set. The result is obtained by surgery.

]]>

Category: Working Seminar on Complex Dynamics

Date: 18.12.19

Time: 11:00 - 12:30

Additional Information:

We study the connectivity of Fatou components for singularly perturbed maps with a McMullen-like Julia set.

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