Wednesday May 17, 14.30h-15.30h (Course on Ergodic Theory and Applications to Complex Dynamics)
Speaker: Anna Jové (UB)
Title: Measure of maximal entropy for Julia sets
Abstract: This is part of the lectures on Ergodic Theory. In this talk, we will discuss the existence of a measure of maximal entropy on the Julia set of a given function, proved by Brolin for polynomials and by Freire-Lopes-Mañé and Lyubich for rational maps.
--------------------
Wednesday May 10, 14.30h-15.30h (Course on Ergodic Theory and Applications to Complex Dynamics)
Speaker: Dan Paraschiv (UB)
Title: An application of ergodic theory in holomorphic dynamics
Abstract: We will construct a rational map R which is ergodic with respect to the Lebesgue measure. Since ergodicity implies topological transitivity, the Julia set of the map R is the entire Riemann sphere. The first such example was given by Lattès. We will present the construction of a so-called Lattès map, and state several of its properties.
--------------------
Wednesday April 19, 14.30h-15.30h (Course on Ergodic Theory and Applications to Complex Dynamics)
Speaker: Anna Jové (UB)
Title: Boundary behaviour of the universal covering
Abstract: In this talk we will discuss the universal covering of a multiply connected domain. In particular, we are interested in the boundary behaviour of such maps and which similarities (and differences) appear compared to the boundary behaviour of the Riemann map. We will answer the following questions:
- When does the radial extension of the universal covering exist?
- Which role play non-contractive curves? (spoiler: non-contractive curves play the same role as crosscuts!)
- Under which conditions does the radial extension of the universal covering induce a measure in the boundary of the domain? What can be said about the support of this measure?
--------------------
Wednesday March 15, 15.15h-16.15
Speaker: Christian Henriksen (Technical University of Denmark)
Title: Dynamics of Chebyshev polynomials.
Abstract: Given a compact, non-polar subset $K$ of the complex plane, one can associate a sequence of suitably normalized Chebyshev polynomials $(T_n)$ to $K$. Each polynomial defines a measure of maximal entropy $\omega_n$, and we show that $\omega_n$ converges weak star to the equilibrium measure $\omega$ of $K$. This result is joint work with C.L. Petersen, H.L. Pedersen and J.S. Christiansen
--------------------
Wednesday March 8, 14.30h-15.30h (Course on Ergodic Theory and Applications to Complex Dynamics)
Speaker: Anna Jové (Universitat de Barcelona)
Title: Inverse branches for inner functions at boundary points
Abstract: Let g be an inner function, and denote by E(g) the set of singularities of g in the unit circle. Hence, we can consider g extended as a meromorphic function in the Riemann sphere minus E(g). In this talk, we will discuss under which conditions inverse of g are well-defined at points in the unit circle. Moreover, we will prove that radial segments are mapped inside Stolz angles of fixed opening, under certain conditions. This gives a generalization of Lemma 2 of [Przytycki+Zdunik, 1994].
--------------------
Wednesday March 1, 14.30h-15.30h (Course on Ergodic Theory and Applications to Complex Dynamics)
Speaker: Toni Garijo (Universitat Rovira i Virgili)
Title: Introduction to Ergodic Theory. Part VII (Entropy)
Abstract: This is the seventh session of a series of seminars about Ergodic Theory. In this talk, we will discuss two different definitions of entropy: the topological entropy and the theoretic entropy. We can also relate both definitions via the variational principle.
-------------------
Wednesday 15 de febrer de 2023, 16:00
Speaker: Kostiantyn Drach ( Institute of Science and Technology Austria (ISTA)).
Title: Universal dynamics in 3D stationary Euler flows
Abstract: In one-dimensional complex dynamics, a branch of dynamical systems that studies iterations of holomorphic maps, the rigidity question has the classical form: Under which conditions can one promote topological conjugacy between a pair of maps to an analytic (conformal) conjugacy? We call this 'parameter rigidity', as it allows us to distinguish maps within parameter space starting with some 'soft' topological (or even combinatorial) data. On the other hand, for a given map, there is a parallel 'dynamical rigidity' question: Can one distinguish individual orbits in combinatorial terms (e.g. via symbolic dynamics)? For polynomials, and especially for quadratic polynomials, this circle of questions is well-studied in the works of Avila, Douady, Hubbard, Lyubich, McMullen, Sullivan, van Strien, Yoccoz, and many others. In this case, the progress was possible thanks to the fact that most polynomials have a naturally defined Markov partition in their dynamical plane (via so-called Yoccoz puzzles). This is not true for general rational maps, and hence life becomes more complicated (or interesting?). In my talk, I will discuss what we know so far about dynamical and parameter rigidity of general rational maps. Our guiding example will be Newton maps, a family of maps naturally arising from Newton's root-finding method. I will also outline a 'toolbox' of techniques useful to attack these types of questions; the key 'tool' is a renormalization concept of complex box mapping. The main message of the talk hopefully will be that life is not that complicated after all (under certain conditions).
-----------------
Wednesday February 8, 14.30h-15.30h
Speaker: Linda Keen (The City University of New York)
Title: Meromorphic functions with an asymptotic polar value
Abstract: The work described in this lecture is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. It includes various joint projects with Tao Chen, Nuria Fagella and Yunping Jiang.
Unlike rational maps, transcendental meromorphic functions have an es- sential singularity, and its preimages cannot be iterated indefinitely. It also means the functions have infinite degree, and if there are only finitely many singular values, there must be asymptotic values — for example, 0 for ez. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two com- plex parameters. Like quadratic polynomials and rational maps of degree 2, understanding their parameter spaces is key to understanding families with more asymptotic values.
Starting in the late ’80’s, the exponential family Ea(z) = exp(z)+a, with asymptotic values at 0 and ∞, was studied by Devaney, Goldberg, Rempe and Schleicher, among others, Both the dynamic and parameter spaces exhibit phenomena not seen for rational maps, for example, “Cantor bouquets”.
The tangent family λtanz, with asymptotic values ±λi is another ex- ample, and, because the asymptotic values are both finite, is more like an “infinite version” of rational maps. Both are one dimensional “slices” of the two dimensional space. Insisting that a fixed point be attractive with a fixed multiplier defines a third example, again more akin to rational maps.
In this lecture, we look at a fourth family in which one of the asymptotic values is a pole, the “polar asymptotic value” of the title. Although these functions can never be hyperbolic, we will show they exhibit behavior we see in both the tangent and exponential families and interpolate between them.
----------------------
Wednesday February 1, 14.30h-15.30h
Speaker: Robert Florido (Universitat de Barcelona)
Title: Mixing. (Course on Ergodic Theory and Applications to Complex Dynamics)
Abstract: This is the sixth session of a series of seminars about Ergodic Theory. In this talk, we will discuss the notion of mixing for measure-preserving transformations.
Thursday November 3, 14.45h-15.45h
Speaker: Anna Jové (Universitat de Barcelona)
Title: Introduction to Ergodic Theory. Part II (Course on Ergodic Theory and Applications to Complex Dynamics)
Abstract: This is the first session of a series of seminars about Ergodic Theory. It is meant to be an introductory session, defining the objectives for the rest of course. Main concepts and theorems will be defined, and appropriate references will be given.
-----------
Thursday October 13, 11.00h-12.00h
Speaker: Michael Yampolsky (University of Toronto)
Title: How to lose at Monte Carlo
Abstract: I will talk about the theoretical challenges to the numerical study of dynamical systems. I will broadly discuss what practitioners attempt to compute, and whether such computations are always possible. Such questions lead to interesting mathematics with surprising practical implications. As an instructive example of the limitations on our ability to compute things, I will describe a “nice” one-dimensional dynamical system for which a numerical approximation of the long-term statistical behavior of the orbits is not possible. In particular, the Monte Carlo simulation probably fails for it.
---------------
Thursday October 6, 14.45h-15.45
Speakers: Anna Jové (UB)
Title: Density of repelling periodic points in the boundary of a basin for rational maps
Abstract: In this talk, we will work on the proof of Przytycki and Zdunik, concerning the density of accessible repelling periodic points in the boundary of a basin of attraction or a parabolic basin for rational maps. We will focus on the particular case of a simply connected basin of attraction. Interesting tools from different fields are used, including ergodic theory, Pesin's theory, and the behaviour of conformal maps and of finite Blaschke products.
-------------------
Thursday September 22 14.45h-15.45h
Speakers: Igsyl Domínguez Calderón (Universidad Catñolica de Chile)
Title: No hiperbolicidad de polinomios fibrados II
Abstract: La conjetura de Fatou, de 1920, sobre polinomios hiperbólicos, ha sido de
gran interés en las últimas décadas aunque sin resultados concluyentes. Buzzard logró probar que esta conjetura no tiene validez en dimensión 2 compleja. En esta charla probaremos que en una “dimensión intermedia” y con herramientas menos
sofisticadas a las usadas por Buzzard, esta conjetura es falsa para polinomios fibrados holomorfos.
--------------------
Thursday September 15 , 14.45h-15.45h
Speakers: Igsyl Domínguez Calderón (Universidad Católica de Chile)
Title: No hiperbolicidad de polinomios fibrados (I)
Abstract: La conjetura de Fatou, de 1920, sobre polinomios hiperbólicos, ha sido de
gran interés en las últimas décadas aunque sin resultados concluyentes. Buzzard logró probar que esta conjetura no tiene validez en dimensión 2 compleja. En esta charla probaremos que en una “dimensión intermedia” y con herramientas menos
sofisticadas a las usadas por Buzzard, esta conjetura es falsa para polinomios fibrados holomorfos.
-------------------
Tuesday June 7, 15.00h-16.00h
Speakers: Jordi Canela (Universitat Jaume I, Castelló)
Title: Dynamics of a connected Julia set which contains a Cantor set of quasicircles III.
Abstract: In this talk we will continue the study of the Chebyshev-Halley family of root finding algorithms from the point of view of singular perturbations. We will focus on the quasiconformal surgery procedure which allows us to relate its dynamics with the one of McMullen's map $z^4+\lambda/z^2$,
--------------------
Monday May 16, 16.00h-17.00h
Speakers: Nuria Fagella (UB)
Title: J-stability for meromorphic maps
--------------------
Date: 25/04/2022, Time: 15:00 (UTC+2) (Joined with GSD-UAB seminar)
Speaker: Xavier Buff (Institut de Mathématiques de Toulouse)
Title: Spiraling domains in dimension 2
Abstract: I will present a work in progress with Jasmin Raissy. We are studying the complex dynamics of the map (x,y) -> (x+y^2+2x^2y,y+x^2+2y^2x). This polynomial map fixes the origin in C^2 and is tangent to the identity at the origin. We are trying to prove that the interior of the basin of attraction of the origin contains infinitely many fixed connected components. Those fixed components are closely related to the periodic trajectories in an equilateral triangular billiard.
------------------
Monday April 4 17.00h-18.00h
Speakers: David Martí-Pete (University of Liverpool)
Title: A counterexample to Eremenko's conjecture
Abstract: For a transcendental entire function, the escaping set
consists of the points that escape to infinity under iteration. This
set was first studied by Eremenko on 1989, who proved that it is
always non-empty and that the boundary of the escaping set is the
Julia set. He also proved that the components of the closure of the
escaping set are all unbounded, and conjectured that, in fact, the
components of the escaping set are all unbounded. The conjecture is
true for several classes of transcendental entire functions, such as
finite order functions in the Eremenko-Lyubich class. Recently,
together with Rempe and Waterman, we constructed a counterexample to
Eremenko's conjecture, and I will sketch this construction.
--------------------
Monday March 14, 17.00h-18.00h
Speakers: Toni Garijo (Universitat Rovira i Virgili)
Title: A Bifurcation analysis of the Microscopic Markov Chain. Approach to contact-based epidemic spreadings
Abstract: We present the first results obtained in the study of the epidemic spreading in a network. This is. joint work with A. Arenas, S. Gómez and J. Villadelprat.
-----------------------
March 7, 16.45h-18.00h
Speakers: Anna Jové (UB)
Title: Indecomposable continua in the boundary of the Baker domains of z+exp(-z)
Abstract: The transcendental entire function z+exp(-z) is well-known for having uncountably many Baker domains of degree 2. However, the boundary of them has been mostly unexplored. In this talk, we will give a wide description of the boundary orbits. More precisely, we will see that all escaping orbits are organized in curves (the so-called hairs or dynamic rays). The accumulation of some of such hairs is a point, while for others is an indecomposable continua. We will give a proof of the existence of such indecomposable continua.
-----------------------
Tuesday January 22, 16.00h-17.00h
Speakers: Dan Paraschiv (UB)
Title: Newton-like components in the Chebyshev-Halley family of degree n polynomials
Abstract: Following Canela, Campos, and Vindel, we study the Cebyshev-Halley methods applied to a family of polynomials. We prove the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. We also prove that, for n>1, the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconformal deformation of the Julia set of the map obtained by applying Newton's method.