## Invited Speakers Include:

**Fatou flowers and parabolic curves**

Marco Abate (U. Pisa)

Abstract: The local dynamics of a one-dimensional holomorphic germ tangent to the identity is described by the classical Leau-Fatou flower theorem, that shows how a pointed neighborhood of the fixed point can be obtained as union of a finite number of forward- or backward-invariant open sets (the petals of the Fatou flower) where the dynamics is conjugated to a translation in a half-plane. In this talk we shall present what is known about generalizations of the Leau-Fatou flower theorem to holomorphic germs tangent to the identity in several complex variables, where the petals are replaced by parabolic curves, starting from the fundamental results by Écalle and Hakim and ending with some very recent developments (work in progress with J. Raissy and T. Servi).

**SRB measures for partially hyperbolic systems whose central direction is weakly expanding**

José Ferreira Alves (U. Porto)

Abstract: We consider diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition $ E^s \oplus E^{cu} $. Assuming the existence of a set of
positive Lebesgue measure on which $ f $ satisfies a weak nonuniform expansivity assumption in the centre unstable direction, we prove that there exists at most a finite number of transitive attractors each of which supports an SRB measure. As part of our argument, we prove that
each attractor admits a Gibbs-Markov-Young geometric structure with integrable return times. This is a joint work with with C. Dias, S. Luzzatto and V. Pinheiro.

**No smooth Julia sets for Henon maps**

Eric Bedford (Indiana Univ.)

Abstract: We will discuss the polynomial automorphisms of C^2 of positive entropy. These are the finite compositions of complex Henon maps. We will show that the forward and backward Julia sets for these maps are never smooth. Thus this family of automorphisms contains no mappings which are analogous to the 1-dimensional case, where there are the special maps z — > z^d and the Tchebyshev maps.

**Stability within holomorphic families of endomorphisms in P^k**

François Berteloot (U. Toulouse)

Abstract: 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.

**Dynamical Manin-Mumford conjecture**

Charles Favre (École Polytechnique)

**Distribution of postcritically finite polynomials**

Thomas Gauthier (U. Picardie Jules Verne)

Abstract: In the moduli space of degree d polynomials, there exists a probability measure which detects the strongest possible bifurcations, the bifurcation measure. C. Favre and I proved that postcritically finite hyperbolic parameters (resp. Misiurewicz postcritically finite) equidistribute this measure when the length of all critical orbit explode (resp. with some restrictions). With G. Vigny, we then gave strengthened versions of both results using different tools. The aim of this talk is to introduce the bifurcation measure, to describe precisely the results and to present an idea of the different tools we used: arithmetic geometry, complex analysis and combinatorial continuity of complex polynomials. We will first begin with the quadratic family z2 + c in which the bifurcation measure is the harmonic measure of the Mandelbrot set and then explain the additional difficulties that arise in higher dimensional parameter spaces.

**Singularities of plane curves and operads**

Étienne Ghys (ENS-Lyon)

**Complex rotation numbers**

Natalya Goncharuk (National research university HSE)

Abstract: Take a cylinder of height h and glue its two borders by an analytic circle diffeomorphism f. We get an elliptic curve. Its modulus $\tau_f(ih)$ is called the complex rotation number of f+ih.

The question is to study the limit behaviour of $\tau_f(ih)$ as the height of the cylinder h tends to zero. It turns out that the answer is related to dynamical properties of f. For the family of circle diffeomorphisms f+a, the limit values of $\tau_{f+a}(ih)$ (as h tends to 0) form a new fract l set "bubbles", a complex analogue to Arnold tongues. I will tell about geometrical structure of "bubbles".

The talk is based on the joint work of X.Buff and myself.

**Algebraic differential equations with single-valued solutions**

Adolfo Guillot (UNAM-Cuernavaca)

Abstract: Multivaluedness is very common within solutions of ordinary differential equations in the complex domain. We will talk about a recent result describing the differential equations given by meromorphic vector fields in algebraic surfaces having at least one single-valued (possibly not entire) solution.

**Spin Calogero-Moser Spaces and Non-commutative Bispectral Problem**

Emil Horozov (U. Sophia)

Abstract: We prove a very general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and q-difference operators with values in all noncommutative algebras. All known bispectral Darboux transformations are special cases of the theorem. Using the methods of quasideterminants and the spectral theory of matrix polynomials, we explicitly classify the set of bispectral Darboux transformations from rank one differential operators and Airy operators with values in matrix algebras. These sets generalize the classical Calogero-Moser spaces and Wilson's adelic Grassmannian.

**Parabolic blow-ups and limits of dynamical systems**

John Hubbard (U. Cornell)

Abstract: The filled Julia set Kc does not depend continuously on c when pc = z^2 + c has a parabolic cycle. We will explain the structure of the set of limits of Kc's, and why these correspond to "enriched dynamical systems" with more generators than the original dynamics. We will also explore the parallel phenomena in the theory of Kleinian groups, called by Thurston "geometric limits".

**Global bifurcation theory in the plane**

Yulij Il'Yashenko (U. Cornell / Independent University Moscow)

Abstract: The first part of the talk is related to the classification of bifurcations in generic k k -parameter families of vector fields in the plane. This theory is at the very beginning of its construction. Majority of statements consists of problems and conjectures.

The second part will be related to the study of the bifurcation diagrams in such families. These diagrams are defined as the sets in the parameter space of the family for which the corresponding vector fields are not structurally stable. Quite unexpected germs of bifurcation diagrams occur even in the two and three parameter families. In four parameter families they are expected to have numeric invariants.

All the necessary definitions will be given.

**Understanding of groups acting on the circle**

Victor Kleptsyn (U. Rennes)

Abstract: I am going to speak on the current state of the project of understanding of groups' actions on the circle (that includes works of S. Alvarez, B. Deroin, D. Filimonov, D. Malicet, C. Menino, A. Navas, M. Triestino).

Studying suciently nice (no-measure-preserving, nitely generated, suciently smooth or even analytic) actions on the circle, one notices that there is a huge dierence in properties between locally discrete and locally non-discrete ones (and that the minimal locally discrete ones are closer in their properties to those possessing an exceptional minimal set, then to those that are locally non-discrete).

Locally non-discrete actions are already quite well understood; well-known arguments of Loray-Rebelo-Nakai-Scherbakov imply the presence of local
ows in their local closure. These elds become a powerful tool in characterization of such actions; for instance, such actions are topologically rigid; there is a recent preprint of Eskif and Rebelo, devoted to their study, to which I refer for more interesting conclusions.

The locally discrete actions in general thus are the most unexplored part. The main conjecture here is that any locally discrete action admits a Markov partition (the latter is understood in a sense, close, but slightly dierent from the one in the works of Cantwell and Conlon). The presence of such a partition gives a very good control on the dynamics on the group.

Our results imply the existence of a Markov partition in almost all the cases of an action of a group (not only pseudogroup) on the circle by analytic (not only smooth) diffeomorphisms; there are only quite special cases that yet are left.

**Lorenz Renormalization**

Marco Martens (SUNY-Stony Brook)

Abstract: Although Lorenz maps are closely related to unimodal interval maps, the infinitely renormalizable Lorenz maps can exhibit quit different behavior than their unimodal counterparts. There are surprising differences in ergodic theory and in the rigidity of such systems.

**Explosion of smoothness for topological conjugates**

Alberto Pinto (U. Porto)

**Some equidistribution results for lattice points in negative curvature**

Mark Pollicott (U. Warwick)

**Dynamics on Wild Character Varieties and Painlevé Equations**

Jean-Pierre Ramis (U. Toulouse)

**Dynamics and holomorphic foliations: perspectives and problems**

Julio Rebelo (U. Toulouse)

**Degenerate singularities of holomorphic vector elds and related problems**

Helena Reis (U. Porto)

**Rational maps of $\CP^2$ with equal dynamical degrees, no invariant foliation, and two distinct measures of maximal entropy**

Roland Roeder (IUPUI)

Abstract: Little is known about the ergodic properties of rational mappings of $\CP^2$ whose first and second dynamical degrees coincide. Suppose $f:\CP^2 \dashrightarrow \CP^2$ is a rational mapping with equal algebraic and topological degrees. We prove that there is an open set of automorphisms $T \in \mathrm{Aut}(\CP^2)$ for which the mapping $f_T:= T \circ f$ is algebraically stable, has equal first and second dynamical degrees, has no invariant foliation, and has two distinct measures of maximal entropy. One of the measures of maximal entropy is of saddle-type and the other is repelling. Moreover, neither is supported on an algebraic curve. As a corollary, we find that for all $T$ outside of a countable union of proper algebraic subsets of $\mathrm{Aut}(\CP^2)$ the mapping $f_T$ is algebraically stable and has no invariant foliation. This is joint work with Jeff Diller.

**Complex reflection in billiards**

Olga Romaskevich (ENS-Lyon / National research university HSE)

Abstract: Consider a billiard in an algebraic curve. Is there a way to define its complex analogue, i.e. a billiard in the complexification of this curve? The answer is yes and I will show how complex reflection is defined and I will promote its usefulness to the study of real billiards. For example, if one considers 3-periodic orbits in an elliptic billiard than the locus of the centres of inscribed circles of the corresponding triangles is an ellipse. This nice geometric fact can be proven by complex reflection methods.

**Unique ergodicity for foliations in P^2 with an invariant curve**

Nessim Sibony (U. Paris Sud)

**Lattices in pseudo-riemannian hyperbolic spaces**

Nicolas Tholozan (U. Luxembourg)

**Analgebraic geometry**

Sergei Yakovenko (Weizmann Institut)

Abstract: Algebraic geometry studies objects defined by polynomial (algebraic) equations. Functions, curves, integral manifolds defined by polynomial differential equations (ordinary or Pfaffian) are generically transcendent and thus often defy the finiteness results so characteristic for the algebraic geometry. In my talk I will attempt to survey a number of results (both recent and not so) showing that to some extent the constructive nature of the algebraic geometry can be carried out across the frontier of algebraicity.

**Renormalization and rotational attractors of two-dimensional dissipative dynamical systems**

Misha Yampolsky (Toronto)

Abstract: We study dissipative rotational attractors in two settings: Siegel disks of H\eacutenon maps and minimal attractors of diffeomorphisms of the annulus. Jointly with D. Gaydashev, we extend renormalization of Siegel maps and critical circle maps to small 2D perturbations, and use renormalization tools to study the geometry of the attractors. In the Siegel case, jointly with D. Gaydashev and R. Radu we prove that for sufficiently dissipative H\eacutenon maps with semi-Siegel points with golden-mean rotation angles, Siegel disks are bounded by (quasi)circles. In the annulus case, jointly with D. Gaydashev, we prove that for bounded type rotation number, critical annulus maps have a minimal attractor which is a C^0, but not smooth, circle – answering a question of E. Pujals.