The backbone of the workshop is a six-lecture series by Arnaud Cheritat, where he introduces and applies techniques of parabolic implosion.
Abstract: In this course we will review the basic invariants of parabolic fixed points of one complex dimensional holomorphic dynamical systems, and describe how the dynamics get perturbed under some type of perturbations (parabolic implosion). We will end the course by giving a few consequences.
Abstract: We will talk about local dynamics of skew-products P with a (non-degenerate) tangent to the identity fixed point at the origin. We will give an explicit sufficient condition on its coefficients for P to have wandering Fatou components. In particular, we will see that the dynamics of quadratic maps of the form \((z,w) \mapsto (z-z^2,w+w^2+bz^2)\) is surprisingly rich: under an explicit arithmetic condition on \(b\), these maps have an infinity of grand orbits of wandering Fatou components, all of which admit non-constant limit maps. The main technical result is a parabolic implosion-type theorem, in which the renormalization limits that appear are different from previously known cases.
Abstract: We discuss the local centralisers of holomorphic maps at parabolic fixed points. We determine the group structure of these centralisers, in general, and obtain them precisely for a number of specific families of maps. We explain how due to a rigidity phenomenon, elements of the local centraliser are of particular form.
Abstract: We construct an example of cubic polynomials which has two distinct critical points and infinitely many cubic-like renormalizations. Such a polynomial is in the same combinatorial class as an infinitely renormalizable unicritical polynomial, and we further prove that there is a continuum in the combinatorial class containing both.
Abstract: Understanding the geometry of The Mandelbrot set, which records dynamical information about every quadratic polynomial, has been a central task in holomorphic dynamics over the past forty years. Near parabolic parameters, the structure of the Mandelbrot set is asymptotically self-similar and resembles a parade of elephants. Near parabolic parameters on these "elephants", the Mandelbrot set is again self-similar and resembles another parade of elephants. This phenomenon repeats infinitely, and we see different parades of elephants at each scale. In this series of two talks, we will explore the implications of controlling the geometry of these elephants. In particular, we will partially answer Milnor's conjecture on the optimality of the Yoccoz inequality, and see potential connections to the local connectivity of the Mandelbrot set.
Abstract: It has been known since work of Fatou from 1926 that the Julia sets of certain entire functions contain arcs to infinity. This phenomenon was investigated further by Devaney and collaborators in the 1980s, who showed the existence of many such curves, now called "Devaney hairs", for large classes of transcendental entire functions. In the dynamically simplest cases - more precisely, when the map is "of disjoint type" - the entire Julia set is an uncountable union of curves to infinity, which are furthermore embedded in the plane in such a manner that the endpoints of all hairs are accessible from the Fatou set.
Here "disjoint type" means that the map is hyperbolic with connected Fatou set. It is natural to ask whether the above properties hold for all disjoint-type entire functions, but it was shown in 2011 by Rottenfußer, Rückert, Schleicher and myself that this is not the case: there is a disjoint-type entire function whose Julia set contains no arc. This suggests the question of studying more closely those functions of disjoint type whose Julia set is a union of arcs; we say that these maps are "criniferous" (from Latin "crinis", hair, and "ferre", to bear). In this talk, I will describe recent work with Leticia Pardo-Simón on this subject. In particular, we show that the endpoints of the hairs of a criniferous disjoint-type function need not be accessible from the Fatou set.
In contrast, this talk will be accessible to those who have no background in transcendental dynamics!
Abstract: For a transcendental entire function, \(f\), the escaping set \(I(f)\) is of great interest. In the 80s, the strong form of Eremenko’s conjecture asked for an arbitrary transcendental entire function \(f\) if every point \(z \in I(f)\) could be joined with infinity by a curve in \(I(f)\). This was shown to not hold in general and that counterexamples to this conjecture whose set of singular values is bounded must be of infinite order. How close to finite order can a counterexample get? This talk will discuss a first answer to this question and the work being done in improving on it.
Abstract: Singular perturbations of rational maps were introduced by McMullen. He studied the rational family \(M_{\lambda,m,n}(z)=z^m + \lambda /z^n\) in order to provide an example of a rational map with a buried Julia component. The corresponding Julia set is a Cantor set of quasicircles.
In some cases, holomorphic families of rational maps \(R_a\) may have degeneracy parameters, i.e.\ parameters for which the degree of the map \(R_a\) decreases. For instance, \(\lambda=0\) is a degeneracy parameter of \(M_{\lambda,m,n}\). In this talk we will explain how singular perturbations can be used to understand the dynamics of families with degeneracy parameters. More specifically, we will show how to relate the dynamics of a family of rational maps \(R_a\) obtained from Chebyshev-Halley root finding algorithms with the dynamics of the family \(M_{\lambda}(z)=z^4 + \lambda /z^2\).
This is a joint work with Antonio Garijo and Pascale Roesch.
Abstract: In this expository talk, we explore some parabolic techniques following the hyperbolic-parabolic deformations by Cui-Tan.
Abstract: Pseudoperiodic functions, in the sense of Arnol’d, are sums of linear transformations and periodic functions. We study a one-parameter family of pseudoperiodic meromorphic Newton methods which are semiconjugated to finite-type maps via the exponential. We investigate the coexistence of wandering domains and attracting basins for suitable parameters in close relation to the orbit of free critical points and the logarithmic lift of periodic Fatou components. (Work in progress with N. Fagella)
Abstract: We study boundary orbits of unbounded invariant Fatou components of transcendental entire functions. The fact that infinity is an essential singularity of the function gives interesting topological and dynamical properties. In particular, we will present new conditions which implies that periodic boundary points and escaping boundary points are dense in the boundary. This is based on joint work in progress with N. Fagella.
Abstract: We study the 1-parameter family of cubic polynomials with 0 a parabolic fixed point of rotation number \(p/q\). Using the technique of dynamical-para puzzles, we show that every parabolic component is a Jordan domain. As a consequence, the connected locus can be decomposed into the union of the central component and all the limbs attached to it. At last we show that the central component is actually a copy of the Julia set of the corresponding quadratic polynomial with parabolic fixed point.