Sessions in 2023-2024

22nd September 2023 Lunch reservations survey	François Bacher Title: Modulus of continuity for the Poincaré metric of a holomorphic foliation and canonical dynamics Abstract : The global dynamics of a holomorphic vector field on a complex manifold $M$ may often usefully be studied through the static properties of the associated foliation. Considered as immersed Riemann surfaces, the leaves of a holomorphic foliation are uniformized by the hyperbolic disk, the complex plane or the Riemann sphere. When all the leaves are hyperbolic, $M$ is endowed with the leafwise Poincaré metric. This allows us to consider some kind of canonical leafwise dynamics with the hyperbolic time. In general, this metric is a priori only semi-continuous. Dinh, Nguyên and Sibony have proved in 2014 that it is Hölder continuous if $M$ is compact and if the foliation is non-singular. This result is essential to show that the entropy of such a foliation is finite. In this talk, we will expose more precisely their result and the technique of their proof. Then, we will present how such techniques can be used to prove a weaker modulus of continuity in the case of non-degenerate singularities. Sahil Gehlawat Title: The leafwise Poincare metric of a singular Riemann surface foliation. Abstract :We consider singular holomorphic foliations $\mathcal{F}$ of dimension 1 on a complex manifold $M$ with all leaves being hyperbolic Riemann surfaces. Consider the Poincare metric $\lambda_{L}$ on each leaf. It is known to vary smoothly along the leaves and conjectured to vary continuously along the transverse directions. There has been a lot of work on achieving this regularity of this leafwise Poincare metric for certain special cases (most of this is for foliations with discrete singular sets). The study of the regularity of this metric is equivalent to studying the Verjovsky's modulus of uniformization map $\eta$, which is a positive map defined away from the singular set $E$ of the foliation $\mathcal{F}$. In this talk, we study this map $\eta$ for foliations without any restriction on the dimension of the singular set. We will give some sufficient conditions for the continuity of the map $\eta$ on the non-singular set, and also its continuous extension on the singular points. By restricting the foliation $\mathcal{F}$ to a domain $U \subset M$, we have the corresponding modulus of uniformization map $\eta_{U}$. We will also talk about the variation of $\eta_{U}$ when the domain $U$ varies in the Hausdorff sense. This is a joint work with Kaushal Verma.
20th October 2023 Lunch reservations survey	Mattias Jonsson: Title: Birational maps with transcendental dynamical degrees. Abstract : To any dominant rational selfmap of a projective variety one can associate a fundamental invariant, the (first) dynamical degree, measuring the algebraic complexity of the map under iteration. In most computable examples, the dynamical degree is an algebraic number. I will report on joint work with J. Bell, J. Diller and H. Krieger, where we construct birational maps of 3-dimensional projective space with a transcendental dynamical degree.
17th November 2023 Lunch reservations survey	Nuria Fagella Title: Grand orbit relations in wandering domains Abstract : We consider dynamical systems generated by the iteration of a complex entire map with an essential singularity at infinity. We say that two points are in the same Grand Orbit, if they are eventually iterated to the same point, even if the number of iterates needed are different for each of them. Grand orbits induce and equivalent relation in the complex plane, where two points belong to the same class if they belong to the same Grand Orbit. Motivated by the work of McMullen and Sullivan in 98 to to study the Teichmuller space of a rational map, we study the nature of Grand Orbit relations in stable components, specially in those specific for entire transcendental maps (Baker and wandering domains). We will show that all grand orbit relations are discrete or all are indiscrete, when considering connected components of the stable set minus the closure of marked points. We will also give an example of a wandering component, on which discrete and indiscrete grand orbits coexist, something that never occurs for stable components of rational maps. In the first part of the talk we will introduce the basics of iteration of transcendental functions, trying to motivate the results which will be presented in the second part. This talk is based in joint work with Christian Henriksen in '06 and '09, and in work in progress with Vasiliki Evdoridou, Lukas Geyer and Leticia Pardo.
15th December 2023 Lunch reservations survey	Seung uk Jang Title: Kummer Rigidity for Hyperkähler Setups Abstract:It is known that, for K3 surfaces, a holomorphic automorphism with positive topological entropy, along with its Green measure in its volume class, forces the base manifold to be a Kummer surface. Furthermore, the automorphism arises from an affine-linear map on the torus. A natural question arises about whether there is a higher-dimensional analogue, particularly for manifolds with structures reasonably similar to those of K3 surfaces. Hyperkähler manifolds are good candidates for such demands, and we will discuss which proof techniques can be generalized from the K3 setups. This generalization requires the manifold to be projective, with the Green measure exactly the same as the volume measure, attributing to a partial generalization. We will also discuss the desired assumption, that the Green measure is *absolutely continuous* with the volume measure. Techniques for the K3 surfaces do generalize to some extent, but there are 'missing links' to spell out the proof per se. Namely, the link lies on the lamination structure of the Green (1,1)-current.
26th January 2024 Lunch reservation survey	Viet-Anh Nguyen Title: The generalized Lelong numbers and intersection theory Abstract: One year ago in this seminar, Lucas Kaufmann gave a stimulating introduction to the theory of tangent currents for positive closed currents, which had been developed by Tien-Cuong Dinh and Nessim Sibony (2012, 2018), as well as its applications in Complex Dynamics. Continuing his task, we present in this talk a development of the theory which provides us the so-called generalized Lelong numbers. This approach is based on some Lelong-Jensen formulas for holomorphic vector bundles. The notion of Lelong number v(T,x) of a positive closed current T at a single point x in an ambient complex manifold X plays a fundamental role in Complex Analysis and Complex Geometry. In 1982 Henri Skoda formulated this notion for the more general class of positive plurisubharmonic currents. More concretely, we introduce a new concept of the generalized Lelong numbers v(T,V,Ω^{(j)}), where T is a positive plurisubharmonic current in X, V is a complex submanifold in X and Ω^{(j)} is a closed (j,j)-form living on V with 0≤ j≤ dim V. The classical case where V is the point x corresponds to dim V=0. Beside the fundamental work of Tien-Cuong Dinh and Nessim Sibony, our present approach is also inspired by the theory of the Lelong number for positive plurisubharmonic currents along a complex linear subspace in \C^n which was developed by Lucia Alessandrini and Giovanni Bassanelli (1996). Next, we study these new numerical values and establish their basic properties. In particular, we obtain geometric characterizations as well as an upper-semicontinuity of the generalized Lelong numbers in the sense of Yum-Tong Siu (1974). When the current T is positive closed, we also establish some links between the generalized Lelong numbers and Dinh-Sibony cohomology classes. Finally, as an application we give an effective condition (in terms of the generalized Lelong numbers) ensuring that m positive closed currents T_1,\ldots,T_m of possibly different bidegrees (p_j,p_j) for 1≤ j≤ m on a compact K\"ahler manifold X are wedgeable in the sense of Dinh-Sibony.
23rd February 2024 Lunch reservation survey	Julio Rebelo Title: Dynamique des groupes d'automorphismes de variétés de caractères et l'équation Painlevé 6 Abstract:Il s'agit d'un travail commun avec Roland Roeder. On discutera de la dynamique du groupe d'automorphismes holomorphes des surfaces cubiques affines de la forme S_{A,B,C,D}={ (x,y,z) \in C^3:x^2+y^2+z^2+xyz=Ax+By+Cz+D} où A, B, C et D sont des paramètres complexes. Ces sont des systèmes dynamiques naturels sur des variétés de caractères et elles décrivent aussi la monodromie (globale) de l'équation Painlevé 6. Nous allons considérer deux dichotomies concernant ces dynamiques, une appelée "Fatou/Julia" (définie de façon analogue aux notions classiques utilisées pour l'iteration d'une fraction rationnelle) et une autre appelée localement discret/localement non-discret qui se voit comme une version non-linéaire de la notion de sous-groupe discret/non-discret d'un groupe de Lie. Les interactions entre ces deux dichotomies permettent de montrer divers résultats sur la dynamique topologique de ces actions. En particulier, on expliquera la co-existence d'ensembles de Fatou (non-vides) et d'ensembles de Julia à interieur non-vide pour un grand ensemble de paramètres.
29th March 2024 Lunch reservation survey	Matteo Ruggiero Title: On the Dynamical Manin-Mumford problem for planar polynomial endomorphisms. Abstract:Let X be a projective variety. The Dynamical Manin-Mumford problem consists in classifying the pairs (Y,f), where Y is a subvariety and f is a polarized endomorphism of X, such that Preper(f|_Y) is Zariski-dense in Y. In a joint work with Romain Dujardin and Charles Favre, we solve this problem when f is a regular endomorphism of P^2 coming from a polynomial endomorphism of C^2 of degree d>=2, under the additional condition that the action of f at the line at infinity has periodic superattracting points. In this talk, we will present this result, and discuss the difficulties we encounter in the superattracting case.
19th April 2024 Lunch reservation survey	Marc Abboud Title: A rigidity result for automorphisms of affine surface. Abstract:An affine surface is a surface given by polynomial equations. An automorphism of an affine surface is a polynomial transformation that preserves the surface and is invertible there. A loxodromic automorphism is an automorphism with first dynamical degree strictly larger than 1. We show the following result: Let X_0 be a normal affine surface over any algebraically closed field, two loxodromic automorphisms of X_0 share a Zariski dense set of periodic points if and only if they have the same periodic points. The proof is based on the understanding of the dynamics "at infinity", to do so we use valuative techniques and then apply equidistribution techniques from arithmetic dynamics following the recent work of Yuan and Zhang. I will discuss examples of affine surfaces, especially the family of Markov surfaces related to the Character variety of the punctured torus. We show for this family of surfaces the stronger rigidity result that two loxodromic automorphism share a Zariski dense set of periodic points if and only if they share a common iterate. This uses techniques from quasi-Fuchsian representation theory.
28th June 2024 Lunch reservation survey	Romain Dujardin Title: Bifurcations homoclines dégénérées en dimension complexe 2. Abstract: La présence de tangences homoclines est la source principale de bifurcations en dynamique 2-dimensionnelle (réelle ou complexe). Lorsqu’on étudie ce phénomène, il est d’usage de supposer que ces tangences sont quadratiques et se déploient à vitesse non-nulle. En adaptant un théorème réel de Takens, nous montrons que toute famille à 1 paramètre de difféomorphismes holomorphes en dimension 2 présentant une bifurcation homocline admet des tangences homoclines quadratiques de vitesse non nulle. En combinant ce résultat avec les résultats annoncés de Avila-Lyubich-Zhang, et d’anciens résultats que j’avais obtenus avec Lyubich, ceci implique que dans l’espace des applications de Hénon suffisamment dissipatives, le lieu de bifurcation est l’adhérence de son intérieur. Nous étudions aussi les bifurcations induites par les familles admettant des tangences persistantes, qui fournissent une approche particulièrement simple au phénomène de Newhouse complexe.

Sessions in 2022-2023

21st September 2022	Matteo Ruggiero: Title : Dynamique locale des germes superattractifs en dimension 2
12th October 2022	Bernard Reinke: Title : Approximation of post-singularly finite entire functions Abstract: Entire functions with finitely many singular values share many dynamical features with polynomials. This is even more the case in the setting of post-singularly finite functions, here is not only a finite set of singular values, but every singular value also has a finite orbit. I will give a brief overview of the dynamics of psf entire functions and discuss to what extent the dynamics of psf entire functions can be approximated by psf polynomials. This is based on joint work with Malavika Mukundan and Nikolai Prochorov. Clément Lefèvre: Title : Approximation of summable rational maps
16th November 2022	Xavier Buff: Title : Infinitesimal Thurston Rigidity and Eigenvalues of the Thurston operator
18th January 2023	Lucas Kaufmann: Title : Densité de courants et applications en dynamique Abstract: Les courants positifs fermés sont des objets fondamentaux en dynamique, géométrie et analyse complexe. En particulier, il est important de comprendre l'intersection (ou produit) de courants. Une approche a ce problème a été introduite par Dinh et Sibony et s’appui sur la notion de courant densité. Dans cet exposé je présenterai la construction de ces courant et ses propriétés. Je présenterai également des applications à l’étude dynamique des laminations et feuilletages.
15th February 2023	Gabriel Vigny: Title : Théorème d'équidistribution à vitesse exponentielle sur l'espace des paramètres
15th March 2023	Martin Leguil: Title : Mesures SRB en dynamique (partiellement) hyperbolique Abstract: un des buts de la théorie ergodique est de décrire le comportement statistique des orbites d’un système dynamique donné. Pour des systèmes présentant une forme d’hyperbolicité, les feuilletages stable/instable s’avèrent être très utiles dans cette étude. Je rappellerai des notions générales autour de ces feuilletages, et comment ces derniers interagissent avec une classe de mesures particulièrement importantes, les mesures SRB, qui permettent de décrire le comportement d’un « gros » ensemble de points. Pour les systèmes partiellement hyperboliques, je mentionnerai une autre classe de mesures, les mesures u-Gibbs, et comment celles-ci permettent d’étudier les mesures SRB. Je parlerai en particulier d’un résultat récent sur la rigidité de ces mesures obtenu avec S. Alvarez, D. Obata et B. Santiago pour une classe de systèmes Anosov du tore de dimension trois.
17th May 2023	Rohini Ramadas: Title : Degenerations and irreducibility problems in rational dynamics Abstract: Per_n is a (nodal) Riemann surface parametrizing degree-2 rational functions with an n-periodic critical point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic critical point (i.e. to the period-n components of the Mandelbrot set). Two long-standing open questions are: (1) Is Per_n connected? (2) Is G_n is irreducible over Q? In the first half of my talk, I will sketch an argument showing that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a smooth point with Q-coordinates on a compactification of Per_n. This smooth Q-rational point represents a special degeneration of degree-2 rational maps, and as such admits an interpretation in terms of tropical geometry. In the second half of my talk, I will go into more details of the construction of the above-mentioned smooth Q-rational point: I will describe joint work with Rob Silversmith, and mention work-in-progress with Xavier Buff, for using compactifications of Hurwitz spaces and tropical geometry in order to investigate degenerations within Per_n.
14th June 2023	Gabriel Vigny and Johan Taflin: Title : Sparsity of post-critically finite maps of P^k

Sessions in 2021-2022

20th October 2021	Charles Favre Title : Distribution of post-critically finite polynomials in the parameter space and dynamical André-Oort conjecture.
17th November 2021	Sébastien Biebler Title : Sur les automorphismes polynomiaux de C^2 avec composante de Fatou errante Abstract: Dans un article récent (2020) en commun avec Pierre Berger, nous montrons qu'il existe un ensemble localement dense d'automorphismes polynomiaux de R^2 ayant un domaine errant s'étendant en une composante de Fatou errante dans C^2. Nous étudions aussi le comportement statistique des orbites des points dans la composante errante et nous montrons que celui-ci est très compliqué à décrire (comportement historique avec émergence super polynomiale). Dans cet exposé, je décrirai les principales étapes de la preuve, qui utilise notamment le phénomène de Newhouse.
2nd February 2022	Romain Dujardin Title : Dynamique des applications de petit degré topologique sur les surfaces. Abstract: Dans cet exposé je passerai en revue un certain nombre de résultats (tous plus ou moins antérieurs à 2010) sur la dynamique des applications birationnelles et de petit degré topologique sur les surfaces kähleriennes compactes. L’objectif est d’expliquer le cheminement qui mène à la construction de “la" mesure d’entropie maximale, qui met en jeu des travaux de nombreux collègues.
16 March 2022	Bertrand Deroin Title : Propriétés statistiques des feuilletages algébriques complexes sur les surfaces Abstract : On survolera les résultats connus sur ce sujet, notamment concernant l'unique ergodicité, la dimension des courants harmoniques, et la géométrie des ensembles limites et de leurs complémentaires. Ce sera l'occasion de parler d'un problème apparaissant naturellement dans l'étude des feuilletages et qui se décline en dynamique itérative sous la forme suivante : est-il possible qu'un Julia strict d'une fraction rationnelle soit localement invariant par la courbe brownienne ?
20th April 2022 (Part 1)	Virgile Tapiero Title : Absolue continuité des applications semi-extrémales Abstract : Si f est une application holomorphe de CP(2) de degré d, alors les deux exposants de sa mesure d’entropie maximale mu sont minorés par (1/2)log d (Briend-Duval). Le cas où les deux exposants sont minimaux a été caractérisé d’un point de vue régulier (mu est absolument continue par rapport à la mesure de Lebesgue) et d’un point de vue géométrique (f est de Lattès) (Berteloot-Dupont-Loeb). Dans cet exposé on étudie les applications possédant un seul exposant minimal. On sait que l’absolue continuité de mu par rapport à la trace du courant de Green implique la minimalité d’un exposant (Dujardin). Nous montrons la réciproque sous certaines hypothèses, en associant des arguments classiques (basés sur des partitions mesurables) à l’existence de formes normales pour les branches inverses de l’application.
20th April 2022 (Part 2)	Florestan Martin-Baillon Title : Courants de bifurcations pour les familles de représentations de groupes en rang supérieur Abstract : Les représentations linéaires agissant sur les espaces projectifs sont des systèmes dynamiques holomorphes qui exhibent une grande variété de comportements. Nous introduirons la notion de stabilité proximal, une généralisation de la stabilité au sens de Sullivan, qui mesure une certaine forme de stabilité dynamique de l’action d’une famille holomorphe de représentations et nous expliquerons comment cette propriété est détectée par un un courant de bifurcation sur l’espace des paramètres de la famille. Ce courant de bifurcation mesure la pluriharmonicité du plus grand exposant de Lyapunov de la famille de représentation, associé à une marche aléatoire.
18th May 2022	Henri de Thélin Title : Courants laminaires et applications