We construct a transcendental entire function for which infinitely many Fatou components share the same boundary. This solves the long-standing open problem whether Lakes of Wada continua can arise in complex dynamics, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. Our theorem also provides the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a recent question of Boc Thaler. Using the same techniques, we give new counterexamples to a conjecture of Eremenko concerning curves in the escaping set of an entire function. This is joint work with Lasse Rempe and James Waterman.[-]

Combination theorems play an important role in several areas of dynamics, geometry, and group theory. In this talk, we will expound a framework to conformally combine Kleinian (reflection) groups and (anti-)holomorphic rational maps in a single dynamical plane. In the anti-holomorphic setting, such hybrid dynamical systems are generated by Schwarz reflection maps arising from univalent rational maps. A crucial technical ingredient of this study is a recently developed David surgery technique that turns hyperbolic conformal dynamical systems to parabolic ones. We will also mention numerous consequences of this theory, including 1. an explicit dynamical connection between various rational Julia and Kleinian limit sets,2. existence of new classes of welding homeomorphisms and conformally removable Julia/limit sets, and3. failure of topological orbit equivalence rigidity for Fuchsian groups acting on the circle.[-]

A transcendental entire function with bounded singular set that is hyperbolic and has a unique Fatou component is said to be of disjoint type. The Julia set of any disjoint-type function of finite order is known to be a collection of curves that escape to infinity and form a Cantor bouquet, i.e., a subset of $\mathbb{C}$ ambiently homeomorphic to a straight brush. We show that there exists $f$ of disjoint type whose Julia set $J(f)$ is a collection of escaping curves, but $J(f)$ is not a Cantor bouquet. On the other hand, we prove that if $f$ of disjoint type and $J(f)$ contains an absorbing Cantor bouquet, that is, a Cantor bouquet to which all escaping points are eventually mapped, then $J(f)$ must be a Cantor bouquet. This is joint work with L. Rempe.[-]

In a recently completed paper Pascale Roesch and I have given a complete proof that the connectedness locus $M_{1}$ in the space moduli space of quadratic rational maps with a parabolic fixed point of multiplier 1 is homeomorphic to the Mandelbrot set. In this talk I will outline and discus the proof, which in an essential way involves puzzles and a theorem on local connectivity of $M_{1}$ at any parameter which is neither renormalizable nor has all fixed points non-repelling similar to Yoccoz celebrated theorem for local connectivity of $M$ at corresponding parameters.[-]

Ecalle's resurgent functions appear naturally as Borel transforms of divergent series like Stirling series, formal solutions of differential equations like Euler series, or formal series associated with many other problems in Analysis and dynamical systems. Resurgence means a certain property of analytic continuation in the Borel plane, whose stability under con- volution (the Borel counterpart of multiplication of formal series) is not obvious. Following the analytic continuation of the convolution of several resurgent functions is indeed a delicate question, but this must be done in an explicit quan- titative way so as to make possible nonlinear resurgent calculus (e.g. to check that resurgent functions are stable under composition or under substitution into a convergent series). This can be done by representing the analytic continuation of the convolution product as the integral of a holomorphic n-form on a singular n-simplex obtained as a suitable explicit deformation of the standard n-simplex. The theory of currents is convenient to deal with such integrals of holomorphic forms, because it allows to content oneself with little regularity: the deformations we use are only Lipschitz continuous, because they are built from the flow of non-autonomous Lipschitz vector fields.[-]

In the recent years, the nature of the generating series of walks in the quarter plane has attracted the attention of many authors in combinatorics and probability. The main questions are: are they algebraic, holonomic (solutions of linear differential equations) or at least hyperalgebraic (solutions of algebraic differential equations)? In this talk, we will show how the nature of the generating function can be approached via the study of a discrete functional equation over a curve E, of genus zero or one. In the first case, the functional equation corresponds to a so called q-difference equation and all the related generating series are differentially transcendental. For the genus one case, the dynamic of the functional equation corresponds to the addition by a given point P of the elliptic curve E. In that situation, one can relate the nature of the generating series to the fact that the point P is of torsion or not.[-]

In this talk we present bifurcation phenomena in natural families of rational or (transcendental) meromorphic functions of finite type $\left\{f_{\lambda}:=\varphi_{\lambda} \circ f_{\lambda_{0}} \circ \psi_{\lambda}^{-1}\right\}_{\lambda \in M}$, where $M$ is a complex connected manifold, $\lambda_{0} \in M, f_{\lambda_{0}}$ is a meromorphic map and $\varphi_{\lambda}$ and $\psi_{\lambda}$ are families of quasiconformal homeomorphisms depending holomorphically on $\lambda$ and with $\psi_{\lambda}(\infty)=\infty$. There are fundamental differences compared to the rational or entire setting due to the presence of poles and therefore of parameters for which singular values are eventually mapped to infinity (singular parameters). Under mild conditions we show that singular (asymptotic) parameters are the endpoint of a curve of parameters for which an attracting cycle progressively exits the domain, while its multiplier tends to zero, proving a conjecture from [Fagella, Keen, 2019]. We also present the connections between cycles exiting the domain, singular parameters, activity of singular orbits and $\mathcal{J}$-unstability, converging to a theorem in the spirit of Mañé-Sad-Sullivan's celebrated result.[-]

Several important problems in complex dynamics are centered around the local connectivity of Julia sets of polynomials and of the Mandelbrot set. Importantly, when the Julia set of a polynomial is locally connected, the topological dynamics ofthe map can be completely described as a quotient of a power map on the circle.Local connectivity of the Julia set is less significant for transcendental entire functions. Nevertheless, by restricting to a class of transcendental entire functions, known as docile functions, we obtain a similar concept by describing the topological dynamics as a quotient of a simpler disjoint-type map. We will discuss the notion ofdocile functions, as well as some of their properties. This is joint work with Lasse Rempe.[-]