I will present work in progress with Jasmin Raissy. We study the dynamics of polynomials maps of C^2 which are tangent to the identity at some fixed point. Our goal is to prove that there exist such maps for which the basin of attraction of the fixed point has infinitely many fixed connected components. This should be the case for the map (x,y)->(x+y^2+2x^2y,y+x^2+2y^2x)

Real quadratic rational maps with real critical points form one of the simplest and most classical dynamical systems. There has been real progress in recent years, especially due to work of Khashayar Filom and Kevin Pilgrim; but there are still open problems.

Joint work with Araceli Bonifant and Scott Sutherland

In this talk, based on an ongoing work with Eric Bedford, Rostislav Grigorchuk and Mikhail Lyubich, I will present how the spectrum of the Laplacian on the Basilica Schreier graphs is related to the iteration of the rational map and to the pullback of a particular line.

The complex rotation number (suggested by V.Arnold) is an invariant related to the dynamics of an analytic circle diffeomorphism $f$. Complex rotation numbers give rise to a nice fractal set "bubbles" analogous to classical Arnold's tongues. 
I will give a survey on complex rotation numbers and list open problems. Also, I will explain how the renormalization operator makes the "bubbles" self-similar and controls their sizes.

Gromov, Dinh and Sibony gave an upper-bound for the topological entropy of any rational map on a projective complex variety. In a joint work with T. T. Truong and J. Xie, we recently extended this bound to arbitrary complete metrized non-Archimedean fields. We also related maps over such fields for which the topological entropy vanishes to the notion of good reduction.

In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen collections of disks. They generalize the classical notion of polynomial-like mapping, as we allow for infinitely many disk components in the domain and finitely many components in the range (while the restriction of the map to each domain component is still a branched covering). In this way, one can talk about generalized renormalization of a given map every time this map restricts to a non-trivial box mapping (even if the map is non-renormalizable in the classical Douady-Hubbard sense). This concept turned out to be extremely useful for tackling diverse problems in natural families of maps.

In our talk, we will discuss rigidity properties of complex box mappings, e.g. quasiconformal rigidity, shrinking of puzzle pieces, etc. This will give us a toolbox of useful results. We will then show how to apply the toolbox almost as a “black box” to conclude similar results in those families of rational maps which are renormalizable in this generalized sense. We will illustrate the success of our strategy with several examples, including polynomials with Siegel disks of bounded type rotation number (work in progress) and Newton maps of polynomials (in both cases, polynomials are of arbitrary degree). The talk is based on joint work with several people, including the participants of the MSRI semester D. Schleicher, S. van Strien, and J. Yang.

In this talk we will explore parameter spaces of meromorphic maps with finitely many singular values. We will list various types of parameters for which the orbits of one or more  singular values have peculiar characteristics, for example, are preperiodic, converge to a parabolic cycle, or are truncated after a certain number of iterates. We will look at the relations between such parameters and show that several of these types of parameters are dense in the bifurcation locus. To do so we will need to implement new strategies with respect to the rational case.   This is joint work with N. Fagella and M. Astorg.

We address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics and goes back to classical work of Fatou and Julia: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This is joint work with Charles Favre and Thomas Gauthier.

We will describe an explicit correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps acting on the Riemann sphere. This correspondence has several dynamical and parameter space consequences. To illustrate these, we will discuss some striking similarities between the deformation spaces of these two classes of conformal dynamical systems, including an analogue of Thurston’s compactness theorem for anti-holomorphic rational maps and relations between the global topology of the corresponding deformation spaces. Based on joint work with Russell Lodge and Yusheng Luo.

Let f:X-->X be a rational map in complex dimension two. Assuming the first dynamical degree of f exceeds 1, there is a standard way to construct a dynamically natural positive closed current invariant under pullback. The construction requires, however, that f be `algebraically stable.' In this talk I'll focus on some recent examples where the dynamical degree is transcendental and construct invariant currents in a situation where algebraic stability is unachievable.

In this talk we will discuss parameter spaces of natural families of transcendental meromorphic maps of finite type,
parametrized over a complex manifold. In this setting, a new type of bifurcation arises for which a periodic cycle can disappear to infinity along a parameter curve.  We shall relate these type of bifurcation parameters, with those for which an asymptotic value is a prepole (virtual cycle parameters), and use these relations to study stability of Julia sets ($J$-stability), concluding that $J-$stable parameters form an open and dense subset  of the parameter space. All our theorems hold for general finite type maps in the sense of Epstein, satisfying certain conditions.
This is joint work with Anna Miriam Benini and Matthieu Astorg.