Category: Working Seminar on Complex Dynamics

Date: 08.04.21

Time: 16:00 - 17:00

Additional Information:

Holomorphic correspondences are polynomial relations $P(z,w)=0$, which can be regarded as multi-valued self-maps of the Riemann sphere (implicit maps

sending $z$ to $w$). The iteration of such a multi-valued map generates a dynamical system on the Riemann sphere (dynamical system which generalises rational maps and finitely generated Kleinian groups). We consider a specific 1-(complex)parameter family of (2:2) correspondences $F_a$ (introduced by S. Bullett and C. Penrose in 1994), which we describe dynamically. In particular, we show that for every a in the connectedness locus $M_{\Gamma}$, this family is a mating between the modular group and rational maps in the family $Per_1(1)$; we develop for this family a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials; and we show that $M_{\Gamma}$ is homeomorphic to the parabolic Mandelbrot set $M_1$. This is joint work with S. Bullett (QMUL)

Category: Working Seminar on Complex Dynamics

Date: 25.03.21

Time: 16:00 - 17:00

Additional Information:

In this talk we will present the concept of \textit{tongue} for families of degree 2 covering maps of the circle. This concept was introduced by Misiurewicz and Rodrigues in analogy to Arnold tongues of orientation preserving homeomorphisms of the circle. Afterwards, we will analyse how these tongues extend beyond their natural range of definition, leading to the concept of \textit{extended tongues}. Finally, we will study the properties of extended tongues for a family of degree 4 Blashcke products which restricts to degree 2 maps of the circle.

]]>Category: Working Seminar on Complex Dynamics

Date: 18.03.21

Time: 16:00 - 17:00

Additional Information:

Category: Working Seminar on Complex Dynamics

Date: 11.03.21

Time: 16:00 - 17:00

Additional Information:

Category: Working Seminar on Complex Dynamics

Date: 25.02.21

Time: 16:00 - 17:00

Additional Information:

It is known that, for many transcendental entire functions with bounded singular set, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are *criniferous*. Although not all functions with bounded singular set are criniferous, those with finite order of growth are, and, in some special cases, their Julia set is a collection of hairs forming a topological object known as *Cantor bouquet*. In this talk, we describe a new class of criniferous functions and explore their relation to Cantor bouquets. This is joint work with L. Rempe.

Category: Working Seminar on Complex Dynamics

Date: 18.02.21

Time: 16:00 - 17:00

Additional Information:

We study the existence of stable manifolds for germs of biholomorphisms in $C^n, n>1$. We will show that if there exists a formal invariant curve for the biholomorphism (a condition that always holds in dimension two) and the restriction to that curve has an attracting behavior (i.e. the multiplier of the restriction is hyperbolic attracting or is a root of unity) then there exist stable manifolds where all the orbits are asymptotic to the formal curve. In the particular case of dimension two, we will show how this result allows to obtain a more general condition for the existence of stable manifolds.

The first part of the talk is based on two joint works with F. Sanz, J. Raissy, J. Ribón and L. Vivas; the second part, on one with R. Rosas.]]>

Category: Working Seminar on Complex Dynamics

Date: 22.12.20

Time: 16:30 - 17:30

Additional Information:

Suppose that $A \subset \C$ is an annulus whose two boundary components $\Gamma, \gamma$ are Jordan curves.

Suppose furthermore that $f$ maps a neighborhood for $\Gamma$ biholomorphically to a neighborhood of $\gamma$,

and that $f(\Gamma)=\gamma$. Then $T=\overline{A} / f$ is a torus, with a distinguished closed curve.

We propose a fast method for numerically computing the complex modulus of $T$ and a related number $\lambda$

which we call the virtual multiplier.

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Category: Working Seminar on Complex Dynamics

Date: 01.12.20

Time: 16:00 - 17:00

Additional Information:

The study of collective synchronization in large ensembles of coupled oscillators has spurred the development of dimensional reduction techniques in nonlinear sciences. We first review the exact method introduced by Watanabe and Strogatz in 1994, which reduces N-dimensional systems of identical phase oscillators to just 3-dimensional ones. Then, we reveal that the conformal automorphisms of the unit disk are actually responsible for such exceptional dynamics. These special Möbius transformations, in close relation with hyperbolic geometry and Poisson kernels, unveil interesting properties of the reduced dynamics. Finally, we simulate the paradigmatic Kuramoto-Sakaguchi model and its extension to two sinusoidally coupled populations, unfolding the so-called chimera states.

]]>Category: Working Seminar on Complex Dynamics

Date: 24.11.20

Time: 16:00 - 17:00

Additional Information:

We determine the Hausdorff and packing dimension of sets of points which escape to

infinity at a given rate under non-autonomous iteration of exponential maps. In particular

we generalize the results proved by Sixsmith in 2016. This is a joint work with Krzysztof Baranski.

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Category: Working Seminar on Complex Dynamics

Date: 10.11.20

Time: 15:30 - 16:30

Additional Information:

In the previous talks, there was given a detailed description of the dynamical plane for the studied family of maps. We proceed with brief proofs of the main theorems of the paper. Theorems A and B deal with achievable connectivities for a small enough perturbation. Theorem C shows that for any possible connectivity, according to Theorem A, there exists a parameter for which the dynamical plane contains a Fatou component of the desired connectivity.

]]>Category: Working Seminar on Complex Dynamics

Date: 27.10.20

Time: 15:30 - 16:30

Additional Information:

Newton's root finding algorithm, called Newton's method, applied to entire or meromorphic functions, has been widely studied in complex dynamics. In 1964 Traub introduced another root finding algorithm, somehow related to Newton's method, known as Traub's method. Motivated by the very interesting dynamical properties of Newton's map we consider a generalised family of maps, which includes Traub's map, and we look at some of its dynamical properties when applied to a polynomial

Category: Working Seminar on Complex Dynamics

Date: 20.10.20

Time: 16:30 - 17:30

Additional Information:

Joint work with Artur Ávila and Xavier Buff. We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indifferent fixed point at 0, to be parameterized by the rotation number α, to depend on α in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional.

]]>Category: Working Seminar on Complex Dynamics

Date: 13.10.20

Time: 16:30 - 17:30

Additional Information:

In this talk we present a family of polynomials of arbitrary degree with a one dimensional (complex) parameter space. In the parameter plane of this family we observe similar structures to the limbs of the Mandelbrot set. We relate both parameter planes.

]]>Category: Working Seminar on Complex Dynamics

Date: 06.10.20

Time: 16:30 - 17:30

Additional Information:

We will present the meaning of the Gauss-Seidelization method and how it can be helpful when studying some concrete cases.

]]>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.

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