Complex dynamics and quasi-conformal geometry.

23 oct. 2017, 09:00 → 25 oct. 2017, 19:05 Europe/Paris

Université d'Angers

Our colleague Tan Lei passed away in April 2016. Her profound, creative mathematics continues to have an impact on geometers and spécialists in complex dynamics. This conference honours her memory.

We have invited a number of specialists in Complex dynamics, to give talks about the latest developments of their field. They all strongly interacted with Tan Lei's work.

Poster Poster of the conference

lundi 23 octobre ¶

09:00 Accueil - café

1 Matings and Thurston obstruction¶

For the matings of quadratic polynomials, the mateability was characterized by Mary Rees and Tan Lei, via Levy cycle theorem. For higher degree polynomials, with the absence of Levy cycle theorems, mateablity criterion is much harder to obtain. In this talk, I will discuss a possible characterization via a tree defined from the Thurston obstruction.

Orateur: Mistuhiro Shishikura (Kyoto University)

Transparents Shishikura-Angers-for-TanLei.pdf

11:00 PAUSE

2 Julia sets with a wandering branching point.¶

According to the Thurston no wandering triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia sets: there exist cubic polynomials whose Julia set is a locally connected dendrite with a branching point which is neither preperiodic nor precritical. We shall reprove this result, constructing such cubic polynomials as limits of cubic polynomials for which one critical point eventually maps to the other critical point which eventually maps to a repelling fixed point. This is a joint work with Jordi Canela and Pascale Roesch.

Orateur: Xavier Buff (Université de Toulouse)

3 The Milnor-Thurston determinant and the Ruelle transfer operator.¶

The topological entropy $\text{Undefined control sequence htop}$ of a continuous piecewise monotone interval map measures the exponential growth in the number of monotonicity intervals for iteratesof the map. Milnor and Thurston showed that $\text{Undefined control sequence htop}$ is the smallest zero of an analytic function, now coined the Milnor-Thurston determinant, that keeps track of relative positions of forward orbits of critical points. On the other hand $\text{Undefined control sequence htop}$ equals the spectral radius of a Ruelle transfer operator $L$, associated with the map. Iterates of $L$ keep track of inverse orbits of the map. For no obvious reason, a Fredholm determinant for the transfer operator has not only the same leading zero as the M-T determinant but all peripheral (those lying in the unit disk) zeros are the same. In the talk I will show that on a suitable function space, the dual of the Ruelle transfer operator has a regularized determinant, identical to the Milnor-Thurston determinant, hereby providing a natural explanation for the above puzzle. This work was inspired by a collaboration with Tan Lei in 2014.

Orateur: Hans Henrik Rugh (Université de Paris-Sud, Orsay)

Transparents Angers-Hans-Henrik.pdf

4 The core entropy for polynomials of higher degree.¶

The notion of topological entropy for real multimodal maps goes back to the work of Milnor and Thurston in the 1970s. In order to extend this definition from the world of real maps to complex polynomials, W. Thurston defined the core entropy as the entropy of the restriction of the polynomial to its Hubbard tree. Together with Tan Lei, her students, and collaborators, a few years ago we set up to understand how this invariant works. In this talk, I will discuss the notion of core entropy and their definition for polynomials of any degree. In particular, we will explore the space PM(d) of "primitive majors" which serves as a combinatorial model for the space of polynomials of degree d, see how to compute the core entropy from the combinatorial data and prove it varies continuously on the parameter space. This is joint work with Gao Yan.

Orateur: Giulio Tiozzo (University of Toronto)

Transparents Tiozzo.pdf

16:45 Pause

5 Desingularizing Hilbert modular varieties.¶

Hirzebruch in the 70’s found a way of resolving the cusps of Hilbert modular surfaces. I will present a new description of this procedure inspired by the dynamics of monomial maps, and show how it extends to Hilbert modular varieties of any dimension.

Orateur: John Hubbard (Cornell University and Université Aix-Marseille)

mardi 24 octobre ¶

6 Generic one parameter perturbation of parabolic points with several petals.¶

With Christiane Rousseau we study generic one-parameter perturbation of holomorphic vector fields in complex dimension one, with the aim of applying this to the study of bifurcation loci of one-parameter families.

Orateur: Arnaud Cheritat

Transparents Cheritat.pdf

10:00 Pause-Café

7 Geometric questions on Julia sets.¶

We will address questions coming from quasiconformal geometry that will be specialized to Julia sets of rational maps.

Orateur: Peter HaissinskY (Universite d'Aix-Marseille)

8 Wandering domains of transcendental functions (joint work with K. Baranski, X. Jarque and B. Karpinska)¶

We present several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of the successive iterates of a wandering component. We shall also construct an oscillating domain in class B on which the iterates are univalent.

Orateur: Nuria Fagella (Universitat de Barcelona)

Transparents Fagella.pdf

9 Two Moduli Spaces.¶

A discussion of two moduli spaces and their awkward topologies: first the space of divisors on the Riemann sphere modulo the action of Moebius automophisms; and second the (compactified) space of curves in the complex projective plane modulo projective automorphisms. This is joint work with Araceli Bonifant.

Orateur: John Milnor (Institute for Mathematical Sciences. Stony Brook)

Transparents Milnor.pdf

10 Computing the conformal dimension of Julia sets by elastic graphs.¶

One measure of the complexity of a Julia set are various notions of "conformal dimension". We show how to estimate the Ahlfors regular conformal dimension sharply from above and below by using energies of maps between graphs, a refinement of the earlier theorem that characterized rational maps using similar energies. This is joint work with Kevin Pilgrim.

Orateur: Dylan Thurston (Indiana University, Bloomington)

Transparents Thurston.pdf

16:45 Pause-Café

11 When hyperbolic maps are matings.¶

A mating is a rational map made by combining two polynomials of the same degree in a certain fashion. Matings were a recurring theme in Tan Lei's work, not surprisingly, since the concept was invented by Douady and Hubbard after their extraordinary success in describing the Mandelbrot set in the parameter space of quadratic polynomials. in fact, Tan Lei's thesis was essentially an existence result, prompted by a question of Douady, and showing that matings are in plentiful supply. It was, however, realised early on that not all rational maps can be described in terms of matings of polynomials. Nevertheless, there are regions of the parameter space of quadratic rational maps in which matings do give a good combinatorial description of the parameter space, and describe all hyperbolic rational maps of bitransitive type. I will talk about a relatively new instance of this, in the case where all Fatou components have disjoint closures.

Orateur: Mary Rees (University of Liverpool)

Transparents Rees.pdf

mercredi 25 octobre ¶

12 On combInaTorIal types of Cycles under zd¶

The talk is based on joint work with Saeed Zakeri. Rotation sets for $z^d$, sets on which $z^d$ is topologically conjugate to a rigid rotation, are well studied in the literature. Much less is known about periodic orbits of other types of combinatorics. To be precise by a combinatorics (of period $q$) we mean a dynamics on $\text{Undefined control sequence TT}$ fixing $0\equiv 1$ and which acts as a permutation of order $q$ on the $x_i$. Which combinatorics are realized under $z^d$? In how many distinct ways is a given combinatorics realized? How does this number depend on the degree $d$?

Orateur: Carsten Lunde Petersen (INM at Roskilde University)

Transparents Petersen.pdf

10:00 Pause-Café

13 Rationality is practically decidable for Nearly Euclidean Thurston maps.¶

A Thurston map $f:(S^2,P)\to (S^2,P)$ is \emph{nearly Euclidean} if its postcritical set $P$ has four points and each branch point is simple. We show that the problem of determining whether $f$ is equivalent to a rational map is algorithmically decidable, and we give a practical implementation of this algorithm. Executable code and data from 50,000 examples is tabulated at \url{https://www.math.vt.edu/netmaps/index.php}. This is joint work with W. Floyd and W. Parry.

Orateur: Kevin Pilgrim (Indiana Universityl. Boomington.)

14 Cubic Polynomials.¶

One of Tan Lei's interest was to understand dynamically the space of cubic polynomials. In this talk we will focus on this question.

Orateur: Pascale ROESCH (Université Paul Sabatier de Toulouse.)