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 Slides
* Research talks
* Introductory talks
Research talks
Escaping Fatou components of transcendental self-maps of the punctured plane
― Topics in Complex Dynamics 2015 ―
Universitat de Barcelona, Barcelona (Spain), November 24, 2015
Abstract: We study the escaping set of transcendental self-maps of the
punctured plane. The orbits of these points accumulate to zero and/or
infinity following what we call essential itineraries. It can be shown
that for every essential itinerary, there are points in the Julia set
that escape following that itinerary. Therefore, it is a natural
question to ask whether there are examples of Fatou components that
escape in each possible way as well. Using approximation theory we are
able to construct functions with wandering domains and Baker domains
that do this. We also study some concrete examples.
Paper: Escaping Fatou components of transcendental self-maps of the punctured plane, in preparation.
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Annular itineraries for \(\mathbb C^*\)
― Topological and Combinatorial Problems in One-dimensional Complex Dynamics ―
Centro di Recerca Matematica Ennio De Giorgi, Pisa (Italy), October 16, 2013
Abstract: We are interested in studying the different rates of escape of points under iteration by transcendental holomorphic self-maps of \(\mathbb C^*\). Using annular covering lemmas we are able to construct different types of orbits, including fast-escaping or arbitrarily slowly escaping points to either 0, infinity or both of them, and points with periodic itineraries as well. These results are analogous to the ones that Phil Rippon and Gwyneth Stallard recently proved for entire functions.
Paper: The escaping set of transcendental self-maps of the punctured plane, to appear in Ergodic Theory and Dynamical Systems (preprint).
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Dynamic Rays for Transcendental Holomorphic Self-maps of \(\mathbb C^*\)
― 18th International Conference on Difference Equations and Applications ―
Casa de Convalescència, Barcelona, July 23, 2012
Abstract: I am interested in the iteration of holomorphic self-maps of the punctured plane \(\mathbb C^*=\mathbb C \setminus \{0\}\) for which both zero and infinity are essential singularities. These maps are of the form \(f(z) = z^n \exp(g(z)+\) \(+h(1/z))\) with \(n \in \mathbb Z\) and \(g(z), h(z)\) nonconstant entire functions. In particular, I would like to understand what is the structure of the escaping set \(I(f)\), the set of points whose orbit accumulates to zero and/or infinity.
In the setting of transcendental entire functions, A. Eremenko conjectured that every \(z \in \mathcal{I}(f)\) could be joined with \(\infty\) by a curve in \(\mathcal{I}(f)\). In analogy to what G. Rottenfußer, J. Rückert, L. Rempe and D. Schleicher proved for functions in class \(\mathcal B\), we show that this property holds for a class of functions whose singular set is bounded away from zero and from infinity and satisfy some technical conditions which are related to the notion of finite order.
Paper: (with Núria Fagella) Dynamic rays of bounded-type transcendental self-maps of the punctured plane, to appear in Discrete and Continuous Dynamical Systems - Series A (preprint).
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Introductory talks (for a wider audience)
Com trobar zeros de funcions? Un problema molt complex! (in catalan)
― Seminari Informal de Matemàtiques de Barcelona ―
Aula IMUB, Facultat de Matemàtiques, Universitat de Barcelona, June 27, 2013
Abstract: Parlarem d'algorismes per buscar zeros de funcions com a sistemes dinàmics en el pla complex. Ens centrarem sobretot en el Mètode de Newton:\(N_f(z) = z-\frac{f(z)}{f'(z)}\). Descriurem les principals propietats tant locals com globals. Si \(P\) és un polinomi, \(N_P\) és una funció racional i cada arrel de \(P\) és un punt fix atractor de \(N_P\). Però, hi ha punts que no convergeixen a cap arrel de \(P\)? Veurem que sorprenentment hi ha una connexió entre els conjunts oberts de punts pels quals el mètode falla i la iteració de polinomis. Podem escollir un conjunt de condicions inicials que ens garanteixin trobar totes les arrels?
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An Introduction to Complex Dynamics
― Scuola Matematica Interuniversitaria ―
Università degli Studi di Perugia, Perugia (Italy), August 24, 2012
Abstract: Complex dynamics concerns the iteration of a holomorphic function \(f\) in a Riemann surface \(S\). When \(f(S)\subseteq S\), this study can be reduced to the cases where \(S\) is the Riemann sphere \(\hat{\mathbb C}=\mathbb C\cup \{\infty\}\) (rational functions), the complex plane \(\mathbb C\) (transcendental entire functions) or \(\mathbb C^*=\mathbb C\setminus\{0\}\). In the last two cases, the presence of essential singularities makes the dynamics more complex: we often find Cantor bouquets, or indecomposable continua. We survey the main results, including the most recent developments due to a better understanding of the escaping set \(I(f)=\{z\in \mathbb C:|f^n(z)| \rightarrow \infty\}\).
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Intoducció a la dinàmica holomorfa transcendent (in catalan)
― Seminari Informal de Matemàtiques de Barcelona ―
Aula IMUB, Facultat de Matemàtiques, Universitat de Barcelona, June 26, 2012
Abstract: La dinàmica complexa estudia la iteració d'una funció holomorfa \(f\) en superfície de Riemann \(S\). Quan \(f(S)\subseteq S\), aquest estudi es redueix als casos en que \(S\) és l'esfera de Riemann \(\hat{\mathbb C}=\mathbb C\cup \{\infty\}\) (funcions racionals), el pla complex \(\mathbb C\) (funcions enteres transcendents) o \(\mathbb C^*=\mathbb C\setminus\{0\}\). En les dues darreres situacions, la presència de singularitats essencials fa que la dinàmica sigui força més complicada: sovint trobem Cantor bouquets (com el de la imatge), o continus indescomposables. Faré un repàs dels resultats principals, incloent els avenços més recents deguts a la millor comprensió de l'escaping set \(I(f)=\{z\in \mathbb C:|f^n(z)| \rightarrow \infty\}\).
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