Charles Favre

Title: Degeneration of rational maps

Abstract: Yusheng Luo constructed non-Archimedean limits associated with a degenerating sequence of rational maps. I shall present an alternative approach which is based on Berkovich theory. This construction is developed in a joint work with Chen Gong.

Kevin Pilgrim

Title: Cylinders for iterated rational maps

Abstract: There is a ‘dictionary’ between fundamental results in the dynamics of iterated rational maps and of Kleinian groups. On the group side, a deep result of W. Thurston gives necessary and sufficient conditions on a (convex compact) Kleinian group for its space of quasiconformal deformations to be non-compact: the quotient hyperbolic 3-manifold should (i) have incompressible boundary, and (ii) be acylindrical. It is tempting to look for analogs in the setting of rational maps. There are clear analogous sufficient conditions for non-compactness coming from ``pinching deformations’’. The combinatorics, however, are far more complicated on the map side, and these are far from necessary. I will give a panoramic survey of the intuition, conjectures, and known results.

Sebastian van Strien

Title: The Thurston algorithm for real entire transcendental post-singularly finite maps

Abstract: Thurston’s characterisation theorem gives a necessary and sufficient condition for when a branched covering map of the sphere (for which the orbits of the branch points have finite cardinality) can be realised by a rational map. In spite of progress, an analogous result for entire maps of the complex plane seem to be not yet known. In this talk, I will discuss a somewhat different approach in the setting of real entire maps whose post-singular set is real and has finite cardinality.

Jan Kiwi (PUC-Chile)

Title: Quadratic Rational Maps and Asymptotics of Transversality

Abstract: In the complex two-dimensional moduli space of quadratic rational maps, it is of interest to study dynamically defined one-dimensional slices. An interesting collection of such slices are the curves formed by maps having a periodic critical point, of a given period. We obtain a formula for the Euler Characteristic of these curves. The formula stems from the study of degenerate holomorphic families of quadratic rational maps and their rescaling limits.

John Smillie

Title: Hyperbolic Henon Maps

Abstract: I would like to recall some classic work of Lipa and Oliva and advertise some more recent work of Thomas Richards. I will make a connection with recent joint work with Yutaka Ishii.

Valentin Huguin

Title: Moduli spaces of polynomial maps and multipliers at small cycles

Abstract: In this talk, I will show that the multipliers at the cycles with periods 1 and 2 provide a good description of the space P_d of polynomial maps of degree d modulo conjugation by affine transformations. More precisely, the elementary symmetric functions of the multipliers at the cycles with periods 1 and 2 induce a finite birational morphism from P_d onto its image. This result arises as a direct consequence of the following two facts: (1) For each integer p > 1, any sequence of complex polynomials of degree d with bounded multipliers at its cycles with period p is necessarily bounded in P_d(C). (2) A generic conjugacy class of complex polynomials of degree d is uniquely determined by its multipliers at its cycles with periods 1 and 2. I will present a quantitative version of the first statement, which also holds over various valued fields of characteristic 0. The second statement proves a conjecture by Hutz and Tepper and strengthens a recent result by Ji and Xie in the polynomial case.

Nikolai Prochorov

Title: Relative version of Thurston's characterization theorem

Abstract: In the 1980s, William Thurston established his renowned topological characterization of postcritically finite rational maps. This result states that any (with a small family of exceptions called flexible Lattès maps) postcritically finite branched cover f, can be realized (i.e., is conjugate up to isotopy relative to its postcritical set P_f) by a postcritically finite rational map if and only if f does not have a Thurston obstruction.

If A is a finite f-invariant set containing the post-critical set P_f, we can ask whether f can be realized relative to the set A. Naturally, the larger the set A, the more refined this notion of realizability becomes. An analog of Thurston's criteria applies in this broader context as well. Furthermore, similar questions arise for Lattès maps, where Thurston's criterion does not work, and in the transcendental settings, where analogs of Thurston's result are known only for very restricted families.

In my talk, I will address a "relative" version of the realizability question: if a Thurston map f can be realized relative to a set B, under what conditions does it fail to be realized relative to a larger set A (B is a subset of A)? Interestingly, there exists a clear criterion (absence of quite a restricted family of Levy cycles), applicable in a very general context, to answer this question. This criterion can be viewed as a relative version of Thurston's theorem (though with an independent proof) and notably applies to all classical (including Lattès) and transcendental Thurston maps in a unified manner.

If time permits, I will also explain how Thurston's theory can be extended to general finite type maps (in the sense of A. Epstein) and how the ideas above can be helpful for transcendental Thurston's theory in general.

Alex Kapiamba

Title: Singular Parabolic Implosion

Abstract: Parabolic implosion is a powerful tool in complex dynamics which describes the perturbation of parabolic fixed points. In this setting, a multiple fixed point splits into several fixed points. In this talk, we will consider some singular perturbations of parabolic fixed points, where the multiple fixed point may split into fixed points and poles and critical points, and see that parabolic implosion can still be recovered. Based on joint work with Xavier Buff and Caroline Davis.

Malavika Mukundan

Title: Marked cycle curves over Per_1(0) and Per_2(0)

Abstract: Given any holomorphic dynamical system with a marked periodic cycle, we may track the changes in the marked cycle under perturbations of the system in some ambient parameter family. This gives rise to a marked cycle curve, which is a branched covering over the parameter family. In this talk, we discuss marked cycle curves of fixed period over two families: quadratic polynomials, and quadratic rational maps with a critical 2-cycle. We shall construct cell-decompositions for these curves entirely based on the combinatorics at certain parameters in the Mandelbrot set. This is joint work with Caroline Davis, Giulio Tiozzo and Daniel Stoll.

Mitsuhiro Shishikura: TBD

Laurent Bartholdi: TBD

Harry Schmidt: TBD

Eriko HironakaTBD