Category: Working Seminar on Complex Dynamics

Date: 16.07.19

Time: 11:30 - 12:30

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We consider the discrete dynamical system $S$ defined on $\mathbb R^2$ given by the secant method applied to a real polynomial $p$. Every simple root $\alpha$ of $p$ has associated its basin of attraction $\mathcal A(\alpha)$ formed by the set of points converging under $S$ towards $\alpha$ and $\mathcal A^*(\alpha)$ its immediate basin of attraction. We investigate the boundary of the immediate basin of attraction of a root of $p$.

]]>Category: Working Seminar on Complex Dynamics

Date: 25.06.19

Time: 10:00 - 11:00

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We present some new results about the dynamical plane of rational Newton's maps by constructing a puzzle structure.

]]>Category: Working Seminar on Complex Dynamics

Date: 17.06.19

Time: 09:30 - 10:30

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Category: Working Seminar on Complex Dynamics

Date: 12.06.19

Time: 17:00 - 18:00

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Category: Working Seminar on Complex Dynamics

Date: 03.06.19

Time: 09:30 - 10:30

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Misieurewicz and Rodrigues introduced the the concept of tongues of the double standard family as open connected sets of parameters for which the double standard map has an attracting cycle (of a prescribed type).

The boundary of a tongue consists of two curves which intersect tangentially on the tip of the tongue. Misieurewicz and Rodrigues described the order of tangency of these curves for a given tongue (called the fixed tongue) and conjectured that this order is general for all tongues of the family. In this talk we will show that this is true by proving that all tongues of the family form regular cusps. The proof relies on the transversality techniques introduced by Adam Epstein.

]]>Category: Working Seminar on Complex Dynamics

Date: 20.05.19

Time: 09:30 - 10:30

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We study the internal dynamics of simply connected, bounded wandering domains of a transcendental entire map f. While the dynamics on periodic Fatou components and on multiply connected wandering domains are well understood, the internal dynamics on simply connected, bounded wandering domains have so far eluded classication. We fill this gap by classifying dynamics on simply connected wandering domains in terms of the hyperbolic distance between iterates and, at the same time, by whether orbits converge to the boundary of the sequence of domains. These classications dene nine possible cases which we show, using approximation theory, that they are realizable. We deduce a general technique for constructing examples of this type.

]]>Category: Working Seminar on Complex Dynamics

Date: 30.04.19

Time: 09:30 - 10:30

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**Abstract**

We investigate the root finding algorithm given by the secant method applied to a real polynomial $

Category: Working Seminar on Complex Dynamics

Date: 11.04.19

Time: 09:00 - 10:00

Additional Information:

In this talk we will present the tuning technique firstly proposed by Adrien Douady. We mainly focus in the tuning between quadratic polynomials.

]]>Category: Working Seminar on Complex Dynamics

Date: 18.03.19

Time: 09:00 - 10:00

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**Abstract**

Many authors have studied sets, associated with the dynamics of a transcendental entire function, which have the topological property of being a spider's web. In this paper we adapt the definition of a spider's web to the punctured plane. We give several characterisations of this topological structure, and study the connection with the usual spider's web in the complex plane. We show that there are many transcendental self-maps of $\mathbb C^*$ for which the Julia set is such a spider's web, and we construct the first example of a transcendental self-map of $\mathbb C^*$ for which the escaping set is such a spider's web. This is a joint work with Vasso Evdoridou and Dave Sixsmith.

]]>Category: Working Seminar on Complex Dynamics

Date: 18.03.19

Time: 10:00 - 11:00

Additional Information:

We show that any dynamics on any discrete planar sequence $S$ can be realized by the postsingular dynamics of some meromorphic function, provided we allow for small perturbations of $S$. This work was influenced by an analogous result of DeMarco, Koch and McMullen for finite $S$ in the rational setting. The proof contains a method for constructing meromorphic functions with good control over both the postsingular set of $f$ and the geometry of $f$, using the Folding Theorem of Bishop and a classical fixpoint theorem. This is joint work with Christopher Bishop.

]]>Category: Working Seminar on Complex Dynamics

Date: 11.03.19

Time: 15:00 - 16:00

Additional Information:

We study the internal dynamics of simply connected, bounded wandering domains of a transcendental entire map f. While the dynamics on periodic Fatou components and on multiply connected wandering domains are well understood, the internal dynamics on simply connected, bounded wandering domains have so far eluded classication. We fill this gap by classifying dynamics on simply connected wandering domains in terms of the hyperbolic distance between iterates and, at the same time, by whether orbits converge to the boundary of the sequence of domains. These classications dene nine possible cases which we show, using approximation theory, that they are realizable. We deduce a general technique for constructing examples of this type.

]]>Category: Working Seminar on Complex Dynamics

Date: 25.02.19

Time: 09:30 - 10:30

Additional Information:

The mating procedure, introduced by A. Douady in 1983, is a tool to construct rational maps from two suitable polynomials. In this talk we review the main properties of the mating between two polynomials. We will follow the paper "On the notion of mating" by C. L. Petersen and D. Meyer published at the Annales de la Faculté des Sciences de Toulouse in 2012.

]]>Category: Working Seminar on Complex Dynamics

Date: 18.02.19

Time: 15:00 - 16:00

Additional Information:

The mating procedure, introduced by A. Douady in 1983, is a tool to construct rational maps from two suitable polynomials. In this talk we review the main properties of the mating between two polynomials. We will follow the paper "On the notion of mating" by C. L. Petersen and D. Meyer published at the Annales de la Faculté des Sciences de Toulouse in 2012.

]]>Category: Working Seminar on Complex Dynamics

Date: 11.02.19

Time: 09:30

Additional Information:

It is known that local connectivity of Julia sets of polynomials has been a key tool to describe the dynamics. The connection is given by the landing of the external rays associated to the immediate basin of attraction of infinity. Also local connectivity has been extensively studied on the rational scenario. For instance one can prove that if the Julia set of a hyperbolic rational map is connected then it is locally connected. To prove this, one uses Whyborn's Theorem.

A natural question arises here: What about local connectivity of transcendental maps? For instance we can prove that if $f$ is a transcendental entire map having an unbounded Fatou component $U$, then the Julia set of $f$ cannot be locally connected. On the contrary we claim (the proof is still work in process) that $N_f(z)=z-\tan(z)$ (the Newton's map of $f(z)=\sin(z)$) has infinitely many unbounded compoments (this we know) and the Julia set is locally connected.

Category: Working Seminar on Complex Dynamics

Date: 05.02.19

Time: 12:00 - 13:00

Additional Information:

It is known that local connectivity of Julia sets of polynomials has been a key tool to describe the dynamics. The connection is given by the landing of the external rays associated to the immediate basin of attraction of infinity. Also local connectivity has been extensively studied on the rational scenario. For instance one can prove that if the Julia set of a hyperbolic rational map is connected then it is locally connected. To prove this, one uses Whyborn's Theorem.

A natural question arises here: What about local connectivity of transcendental maps? For instance we can prove that if $f$ is a transcendental entire map having an unbounded Fatou component $U$, then the Julia set of $f$ cannot be locally connected. On the contrary we claim (the proof is still work in process) that $N_f(z)=z-\tan(z)$ (the Newton's map of $f(z)=\sin(z)$) has infinitely many unbounded compoments (this we know) and the Julia set is locally connected.

Category: Working Seminar on Complex Dynamics

Date: 30.01.19

Time: 10:30 - 11:30

Additional Information:

It is known that local connectivity of Julia sets of polynomials has been a key tool to describe the dynamics. The connection is given by the landing of the external rays associated to the immediate basin of attraction of infinity. Also local connectivity has been extensively studied on the rational scenario. For instance one can prove that if the Julia set of a hyperbolic rational map is connected then it is locally connected. To prove this, one uses Whyborn's Theorem.

A natural question arises here: What about local connectivity of transcendental maps? For instance we can prove that if $f$ is a transcendental entire map having an unbounded Fatou component $U$, then the Julia set of $f$ cannot be locally connected. On the contrary we claim (the proof is still work in process) that $N_f(z)=z-\tan(z)$ (the Newton's map of $f(z)=\sin(z)$) has infinitely many unbounded compoments (this we know) and the Julia set is locally connected.

Category: Working Seminar on Complex Dynamics

Date: 25.01.19

Time: 12:00 - 13:00

Additional Information:

A natural question arises here: What about local connectivity of transcendental maps? For instance we can prove that if $f$ is a transcendental entire map having an unbounded Fatou component $U$, then the Julia set of $f$ cannot be locally connected. On the contrary we claim (the proof is still work in process) that $N_f(z)=z-\tan(z)$ (the Newton's map of $f(z)=\sin(z)$) has infinitely many unbounded compoments (this we know) and the Julia set is locally connected.

Category: Working Seminar on Complex Dynamics

Date: 09.01.19

Time: 12:00 - 13:00

Additional Information:

The mating procedure, introduced by A. Douady in 1983, is a tool to construct rational maps from two suitable polynomials. In this talk we review the main properties of the mating between two polynomials. We will follow the paper "On the notion of mating" by C. L. Petersen and D. Meyer published at the Annales de la Faculté des Sciences de Toulouse in 2012.

]]>Category: Working Seminar on Complex Dynamics

Date: 19.12.18

Time: 09:00 - 10:00

Additional Information:

Category: Working Seminar on Complex Dynamics

Date: 12.12.18

Time: 12:00 - 13:00

Additional Information: