Parameter Problems in Analytic Dynamics

A conference at Imperial College London

June 27-July 1, 2016

Celebrating Sebastian van Strien's 60th birthday

Schedule of the talks

  MondayTuesdayWednesdayThursday Friday
09:30-10:30 Registration Walter Bergweiler Romain Dujardin Hiroyuki Inou Edson de Faria
11:00-12:00 Mario Bonk Anna Benini Nuria Fagella Misha Yampolsky Genadi Levin
12:00-14:00 Lunch Lunch Lunch Lunch Lunch
14:00-15:00 John Smillie Juan Rivera-Letelier Pascale Roesch Denis Gaidashev Neil Dobbs
15:00-15:30 tea/coffeetea/coffeetea/coffee Free afternoon tea/coffee
15:30-16:30 Eric Bedford Charles Favre Xavier Buff Free afternoon Pablo Guarino
16:30-17:00 tea/coffeetea/coffeetea/coffee Free afternoon tea/coffee
17:00-18:00 Marco Abate Francois Berteloot   Free afternoon Marco Martens

The talks on Monday, Tuesday, and Wednesday will be in Room Huxley 340. The talks on Thursday and Friday will be in Room Huxley 311.

Conference dinner

The conferecne dinner will be on Wednesday from 7:00pm to 11:00pm on a cruise boat along River Thames in London.

We plan to leave the college at 5:45 pm and walk to Westminster Pier. The walk will take an hour and will be mostly through Hyde park, Geen park, and St James's park. Alternatively, one can take the tube from South Kensington tube station to the Westminster tube station, and then walk to the Westminster Pier. The latter trip takes about 20 minutes. Please note that the pier is booked only for a limited time, so it is necessary that you make it to the pier at 6:45pm.

Titles and abstracts of the talks

Mario Bonk, Expanding Thurston maps, slides
Date and time: Monday 11:00 - 12:00
Abstract of the talk: A branched covering map on a 2-sphere S2 is a continuous map that is locally modeled on a rational map on the Riemann sphere. A critical point of such a map f is a point in S2 where f is not a local homeomorphism. Thurston considered branched covering maps for which the forward orbit of each critical point under iteration is finite. These maps are now called Thurston maps.

In joint work with Daniel Meyer we considered Thurston maps that are expanding in a suitable sense. The study of these maps links diverse areas such as dynamical systems, classical conformal analysis, hyperbolic geometry, geometric group theory, and analysis on metric spaces. In my talk I will give a survey on this subject.

John Smillie, The Dynamics of Complex Henon Diffeomorphisms, slides
Date and time: Monday 14:00 - 15:00
Abstract of the talk: I will begin this talk with a general introduction to the dynamics of the complex Henon family of diffeomorphisms. I will ask what might we learn by studying them and describe techniques that have proven effective. Following this I will discuss joint work with Eric Bedford which addresses the question of what is the natural two dimensional analogue of the Misiurewicz property for polynomial maps. This work addresses the connection between regularity of stable and unstable manifolds and uniformity of expansion and the presence of tangencies.

Eric Bedford, No smooth Julia sets for complex Henon maps, slides
Date and time: Monday 15:30 - 16:30
Abstract of the talk: We will give a survey of the Fatou and Julia sets of complex Henon maps. Then we will show that the Julia set is never smooth. This is joint work with Kyounghee Kim.

Marco Abate, Dynamics of families of maps tangent to the identity, slides
Date and time: Monday 17:00 - 18:00
Abstract of the talk: We shall show how it is possible to associate to any holomorphic germ f tangent to the identity in several complex variables a singular holomorphic foliation in Riemann surfaces of a projective space and two meromorphic connections along the foliation, sharing the same (real) geodesics. When f is the time-1 map of a homogeneous vector field, this geodesic flow allows to recover completely the dynamics of the original germ, and can be studied geometrically. In particular, we shall present local normal forms around the singularities and a global Poincare-Bendixson-like theorem giving the asymptotic behavior of geodesics. As a consequence we shall be able to describe the dynamics of some interesting families of germs tangent to the identity in two complex variables in a full neighborhood of the origin. (Joint work with F. Tovena and F. Bianchi).

Walter Bergweiler, Lyapunov exponents and related concepts for entire functions, slides
Date and time: Tuesday 9:30 - 10:30
Abstract of the talk: Let   f   be an entire function and denote by   f #   be the spherical derivative of   f   and by   f n   the n-th iterate of  f . We study how fast   (fn)#(z)   can tend to infinity for a point   z   in the Julia set of   f. We also consider the growth rate of   sup z ∈ U (fn)#(z)   for open sets   U   intersecting the Julia set.

The results are joint work with Xiao Yao and Jianhua Zheng.

Anna Benini, A landing theorem for hairs and dreadlocks of entire functions with bounded post-singular sets
Date and time: Tuesday 11:00 - 12:00
Abstract of the talk: The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the successful study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray.

We prove an analogue of the theorem for entire functions with bounded postsingular set. If such f additionally has finite order of growth, then our result states precisely that every periodic hair of f lands at a repelling or parabolic point, and again conversely every repelling or parabolic point is the landing point of at least one periodic hair. (Here a periodic hair is a curve consisting of escaping points of f that is invariant under an iterate of f .) For general f with bounded postsingular set, but not necessarily of finite order, the role of hairs is taken by more general connected sets of escaping points, which we call dreadlocks . This is joint work with Lasse Rempe-Gillen. Part of these results have been proven independently by M. Rothgang and D. Schleicher.

Juan Rivera-Letelier, Equidistribution in arithmetic geometry and dynamics, slides
Date and time: Tuesday 14:00 - 15:00
Abstract of the talk: A mainly expository account of equidstribution results in arithmetic geometry and dynamics. The focus will be on the celebrated result of Szpiro, Ullmo, and Zhang, and its applications to arithmetic dynamical systems. Alhtough this result has been widely generalized, all of its extensions are restricted to height functions that are extremal in a precise sense. We will show that the equidistribution property can fail dramatically for non-extremal height functions, and give a complete characterization of those toric heights that satisfy the equidistribution property.

Charles Favre, Distribution of PCF cubic polynomials (joint work with Thomas Gauthier), slides
Date and time: Tuesday 15:30 - 16:30
Abstract of the talk: We shall review recent results on the repartition of post-critically finite polynomials in the space of cubic polynomials.

Francois Berteloot, Stability within holomorphic families of endomorphisms in Pk
Date and time: Tuesday 17:00 - 18:00
Abstract of the talk: We extend to the higher dimensional setting the Mane-Sad-Sullivan theorem which gives various characterizations of the stability of holomorphic families of rational maps on the Riemann sphere. This is a joint work with F. Bianchi and C. Dupont.

Romain Dujardin, Non-density of stability for holomorphic mappings on, slides Pk
Date and time: Wednesday 9:30 - 10:30
Abstract of the talk: The stability/bifurcation theory of rational mappings in complex dimension 1 was designed by Mañé-Sad-Sullivan and Lyubich in the early 1980’s. A natural generalization to higher dimensions was recently put forward by Berteloot, Bianchi and Dupont. A salient feature of dimension 1 is that structural stability on J is dense in any holomorphic family of rational maps.

In this talk I will report on work in progress showing that the corresponding result fails in higher dimension, and discuss possible mechanisms leading to robust bifurcations in this context.

Nuria Fagella, Connectivity of Julia sets of Newton maps: A unified approach, slides (joint with K.Baranski, B.Karpinska and X.Jarque)
Date and time: Wednesday 11:00 - 12:00
Abstract of the talk: In this talk I give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than one or an entire transcendental function) is connected. The result is as a consequence of a more general theorem whose proof spreads among many papers, which consider separately a number of particular cases for rational and transcendental maps, and use a variety of techniques. In this talk we present a unified, direct and reasonably self-contained proof which works for all situations alike.

Pascale Roesch, Streching rays
Date and time: Wednesday 14:00 - 15:00
Abstract of the talk: This is a commun work with Shizuo Nakane. We study the landing and non landing of certain streching rays in the family of cubic polynomials.

Xavier Buff, Double parabolic renormalization in the quadratic family, slides
Date and time: Wednesday 15:30 - 16:30
Abstract of the talk: The quadratic rational map   g a (z) = z/(1+ az + z2)   has a double fixed point at 0, except for  a=0  for which 0 is a triple fixed point. We study the behavior of horn maps  ha  associated to the quadratic rational maps  ga  as a tends to 0.

Hiroyuki Inou, On accessibility of hyperbolic components of the tricorn, slides
Date and time: Thursday 9:30 - 10:30
Abstract of the talk: As opposed to the fact that any hyperbolic component of the Mandelbrot set is accessible by external rays, numerical experiments suggest that many hyperbolic components of the tricorn of odd period are not accessible from the escape locus. We discuss existence of inaccessible hyperbolic components of the tricorn by investigating the parameter space of geometric limits and Julia-Lavaurs sets.
This is a joint work in progress with Sabyasachi Mukherjee (Stony Brook University).

Michael Yampolsky, On renormalization of critical circle maps and applications to 2D dissipative dynamics
Date and time: Thursday 11:00 - 12:00
Abstract of the talk: I will review some new results on renormalization of critical circle maps, and will present applications to 2D dissipative dynamics, settling a conjecture of E. Pujals. I will also discuss some open questions and conjectures.

Denis Gaidashev, Universality for the golden mean Siegel disks, and existence of Siegel cylinders, slides
Date and time: Thursday 14:00 - 15:00
Abstract of the talk: The golden mean Siegel disk universality conjecture for analytic maps with a golden mean irrationally neutral fixed point - independence of small scale geometry of a Siegel disk on the particular form of a map - has been one of the long-standing open questions in complex dynamics.

We describe our recent computer-assisted proof of this conjecture, and extend renormalization to the dissipative two-dimensional perturbations of maps with Siegel disks. We prove that renormalization hyperbolicity persists in two-dimensions, and conclude with several consequences such as existence of Siegel cylinders and "warped" Siegel disk attractors for two dimensional dissipative maps.

This is a joint work with M. Yampolsky and R. Radu.

Edson de Faria, Quasi-symmetric rigidity of multicritical circle maps
Date and time: Friday 9:30 - 10:30
Abstract of the talk:The rigidity problem for one-dimensional dynamical systems has been the subject of intense investigation in recent years. In particular, for smooth homeomorphisms of the circle having exactly one critical point of non-flat type, a fairly complete theory has emerged. By contrast, the corresponding theory for circle homeomorphisms having two or more critical points is very far from being well-developed. In this talk I shall present a first step in this direction, in the form of a pre-rigidity result stating that, under certain natural hypotheses, any two such maps are quasi-symmetrically conjugate as soon as they are topologically equivalent by a conjugacy that maps critical points to critical points.

Genadi Levin, On hyperbolic sets of polynomials
Date and time: Friday 11:00 - 12:00
Abstract of the talk: We prove that the Julia set of a quadratic polynomial without irrational cycles is locally connected at the points of hyperbolic sets. We discuss also related problems. This is a work in progress with Feliks Przytycki and Weixiao Shen.

Neil Dobbs, Diabolical entropy, slides
Date and time: Friday 14:00 - 15:00
Abstract of the talk: Milnor and Thurston proved that topological entropy as a function of parameter in the quadratic family is a monotone function. Guckhenheimer showed that it is Hölder continuous. In joint work with Nicolae Mihalache, we provide precise estimates on the Hölder exponent at almost every parameter.

Pablo Guarino, Absolutely continuous measures for bimodal maps
Date and time: Friday 15:30 - 16:30
Abstract of the talk:We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one C2 endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Those conditions are satisfied when the boundary points of the rotation interval belong to a Diophantine class. In particular they hold for Lebesgue almost every rotation interval. The measure obtained is a global physical measure, and it is hyperbolic. Joint work with Sylvain Crovisier (Université Paris-Sud, Orsay) and Liviana Palmisano (Polish Academy of Sciences, Warsaw), available at arxiv:1601.06807.

Marco Martens
Date and time: Friday 17:00 - 18:00
Abstract of the talk: