Abstracts

TALKS

Matthieu Astorg: Local dynamics of skew-products tangent to the identity (joint work with Luka Boc Thaler).

The results we will present in this talk deal with local dynamics of skew-products P with a (non-degenerate) tangent to the identity fixed point at the origin. We will give an explicit sufficient condition on its coefficients for P to have wandering Fatou components. In particular, we will see that the dynamics of quadratic maps of the form (z,w)-> (z-z2,w+w2+bz2) is surprisingly rich: under an explicit arithmetic condition on b, these maps have an infinity of grand orbits of wandering Fatou components, all of which admit non-constant limit maps. The main technical result is a parabolic implosion-type theorem, in which the renormalization limits that appear are different from previously known cases.

Anna Miriam Benini: Bifurcations in families of meromorphic maps.

We will discuss natural parameter spaces of transcendental meromorphic maps with finitely many singular values.
Due to the presence of both poles and asymptotic values, a new type of bifurcation arises for which a periodic cycle can disappear to infinity along a parameter curve. By studying this new type of bifurcations we are able to connect absence of bifurcations to stability of Julia sets, concluding that J-stable parameters form an open and dense subset of the parameter space, in analogy to the celebrated results by Mañé-Sad-Sullivan and Lyubich. All our theorems hold for general finite type maps in the sense of Epstein, satisfying certain conditions. This is joint work with Nuria Fagella and Matthieu Astorg.

Eric Bedford: Survey of the dynamics of complex Henon maps.

This talk will survey the dynamics of complex Henon maps.

François Berteloot: Post-critical normality and stability in holomorphic dynamics.

The stability of holomorphic families of rational functions is characterized by the post-critical normality and this is a fundamental feature of the theory of Mané-Sad-Sullivan and Lyubich.
We will discuss how this works for families of endomorphisms of $P^k$.
This is a joint work with Maxence Brevard.

Gautam Bharali: Unbounded visibility domains and the end compactification.

In the last few years, various inter-related notions of visibility for the Kobayashi distance have appeared in the literature and have proven to be useful in establishing new results on holomorphic maps between domains. Visibility in the Riemannian category is an outcome of negative curvature and thus, for the Kobayashi distance, could be viewed as a weak notion of negative curvature in the complex setting. All earlier works concerning visibility for the Kobayashi distance focused on bounded domains. In this talk, the ideas hinted at above will be extended to unbounded domains. In doing so, we shall consider two slightly different notions of visibility. The weaker of the two visibility properties is good enough to yield a Wolff-Denjoy theorem on a range of domains -- not necessarily bounded -- with quite irregular boundaries. I shall discuss this in passing, but the talk will focus on what exactly the two separate notions of visibility signify, and on examples. This is joint work with Andrew Zimmer.

Sebastien Biebler: Newhouse phenomenon: prevalence and applications.

In the 60s, in a mathematical optimistic movement aiming to describe a typical dynamical system, Smale conjectured the density of uniform hyperbolicity in the space of C^r-diffeomorphisms f of a compact manifold M. In the 70s, Newhouse discovered an extremely complicated new phenomenon, resulting in an obstruction to Smale's conjecture.

Specifically, he showed the existence of (nonempty) open sets U of C^2-diffeomorphisms of a surface M such that a generic map f in U has infinitely many attracting periodic points. In this talk, I will review some applications of this phenomenon, in particular in the holomorphic context, and I will discuss the question of its prevalence.

This talk is based on joint works with Pierre Berger.

Filippo Bracci: Unexpected applications of commuting holomorphic maps’ properties.

Commuting holomorphic maps have been studied since Shields, Behan, Cowen, Abate, Vigué, the speaker, etc, with the main aim of proving existence of common fixed points. In this talk I will revise some results and show how, rather unexpectedly, properties of commuting holomorphic maps can be used to prove results of continuous extension to the boundary of biholomorphisms in convex domains.

Xavier Buff: Spiralling Domains in Dimension 2.

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+x(x-y),y+x^2+y(x-y)).

Davoud Cheraghi: Local centralisers at parabolic fixed points.

We discuss the local centralisers of holomorphic maps at parabolic fixed points. We determine the group structure of these centralisers, in general, and obtain them precisely for a number of specific families of amps. We explain how due to rigidity phenomenon, elements of the local centraliser are of particular form.

Núria Fagella: The Denjoy-Wolff set: from holomorphic sequences to wandering domains.

The dynamics inside periodic components of the stable set has a strong link with classical theorems of complex analysis like the Denjoy-Wolff Theorem about analytic maps of the unit disk. The fractal boundaries of such components arising so naturally from iteration often present interesting topological properties which may play a role when trying to transfer results from the unit disk back to the dynamical plane. However, if the components are not periodic but wandering, we need to reach further and consider non-autonomous iteration. Starting from periodic components, I aim to present some recent results about the dynamics inside wandering domains and also on their boundaries. Many of the results are proven in the very general setting of non-autonomous dynamics or even for sequences of holomorphic maps.

Matteo Fiacchi: Horospheres in several complex variables and applications to dynamics.

In the unit disk of the complex plane the horocycles play an important role in geometric function theory and in complex dynamics. The horospheres are the natural generalisation in several complex variables. Abate in 1988 proved the existence of horospheres in bounded strongly convex domains in higher dimension using Lempert’s theory of complex geodesics. It is difficult to generalize such proof to other classes of domains because Lempert’s theory works well only in the convex case.

In this talk we prove the existence of horospheres on strongly pseudoconvex domains and convex finite type domains, using a metric geometry approach. We introduce a condition, the approaching geodesics property, on a proper geodesic Gromov hyperbolic metric space, which implies that the horofunction compactification is topologically equivalent to the Gromov compactification. Using scaling techniques, we prove the approaching geodesics property for strongly pseudoconvex domains and a weaker version for convex finite type domains.

As a consequence we prove that on those domains big and small horospheres as defined by Abate coincide.

In the last part we apply our results to the study of the forward and backward dynamics of non-expanding self-maps of a proper geodesic Gromov metric space, obtaining in particular a generalization of the classical Julia's lemma when the horofunction and Gromov compactifications are equivalent.

This is a joint work with L. Arosio, S. Gontard and L. Guerini.

Aleksandra Huczek: Some generalizations of the Wolff-Denjoy type theorem.

Nonexpansive mappings i.e. 1-Lipschitz, similarly like isometries and contractions, form one of the most basic class of nonlinear mappings. Currently interesting and at the same time very intriguing is considering dynamics of such mappings. One of theorem which relates to dynamics of nonexpansive mappings is the Wolff-Denjoy theorem. In classical version it describes dynamics of holomorphic self-mappings in respect to the Poincaré distance [7]. In other words, if f : ∆ → ∆ is a holomorphic mapping of the unit disc ∆ ⊂ C without a fixed point, then there is a point ξ ∈ ∂∆ such that the iterates f^n converge locally uniformly to ξ on ∆. The above theorem has been generalized over the years in different directions [1, 2, 4, 6]. The aim of this talk is present some generalizations of the Wolff-Denjoy type theorem in geodesic spaces. We show that if S = {f_t: Y → Y, t≥0} is a one-parameter continuous semigroup of nonexspansive mappings acting on a complete locally compact geodesic space (Y, d) then there exists ξ ∈ ∂Y such that S converge uniformly on bounded sets of Y to ξ. In particular, our result applies to strictly convex bounded domains in R^n or C^n with respect to different metrics [3].

References
[1] M. Abate, J. Raissy, ’Wolff–Denjoy theorems in nonsmooth convex domains’, Ann. Mat. Pura Appl. 193 (2014), 1503–1518.
[2] A.F. Beardon, The dynamics of contractions and analytic maps, J. London Math. Soc., (1990), 41, 141–150.
[3] M. Budzy ́nska, T. Kuczumow and S. Reich, ’Theorems of Denjoy–Wolff type’, Ann. Mat. Pura Appl. 192 (2013), 621–648.
[4] A. Huczek, A. Wi ́snicki Wolff–Denjoy theorems in geodesic spaces, Bull. London Math. Soc., (2021), 53, 1139-1158.
[5] A. Huczek, A. Wi ́snicki Theorems of Wolff–Denjoy type for semigroups of nonexpansive mappings in geodesic spaces, Math. Nachr., (2022), [online].
[6] B. Lemmens at al., Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces, J. Anal. Math., (2018), 134, 671–718.
[7] J. Wolff, Sur une g ́en ́eralisation dun the ́eor`eme de Schwarz, C.R. Acad. Sc. Paris, (1926), 182, 918–920.

Valentin Huguin: Dynamics of Koch postcritically finite endomorphisms.

Many known examples of postcritically finite endomorphisms of projective spaces of dimension at least 2 can be obtained from a construction due to Koch, which is related to Thurston’s topological characterization of postcritically finite rational maps in dimension 1. In this talk, I will examine these endomorphisms, mainly on their postcritical set. In particular, I will give information about their multipliers at their periodic points and explain that each of them either is zero or has modulus greater than 1.

Han Peters: Chaos and phase transitions on regular lattices.

Partition functions on graphs that are in some sense recursively connected often naturally leads to holomorphic dynamical systems. An elegant example was given in recent work of Ombra and Riveira-Letelier, where they study the partition function of the hard core model on Cayley trees. This setting induces the iteration of a one-parameter family of rational functions. In fact, the zero locus of the partition functions coincides with the bifurcation locus of the associated family of rational maps. As a consequence it was shown that there exists a single phase transition of infinite order.

In recent works with de Boer, Buys, Guerini and Regts, we demonstrated that this connection between the zero locus of partition functions and the bifurcation locus of associated rational functions persists in settings that have no clear dynamical interpretation. Examples are the family of all bounded degree graphs, for both the hard-core and the Ising model. Again the zero locus coincides with the interpretation of the bifurcation locus.

In current work we focus on the setting that is most interesting from a physical perspective: graphs converging to a regular lattice. While there is no clear interpretation as a holomorphic dynamical systems, both simulations and preliminary results demonstrate the potential of methods from complex dynamical systems in this setting..

Sergei Yakovenko: On Fuchsian equations with a small parameter before the highest derivative.

Parametric families of ordinary differential equations with a small parameter occurring before the highest derivative are notorious for their malicious behavior. Generically referred to as "singular perturbations", they exhibit all kinds of complicated behavior, both analytically and dynamically: canard limit cycles, exponential and super-exponential asymptotics, ultrafast oscillations, divergence of different formal analytic procedures e.a. Many methods were developed, both by classics and more recently, to deal with such challenges.

However, for some equations we have an a priori qualitative information that their solutions exhibit relatively tame behavior despite the fact that formally the equations "look dangerously". In my talk I will explain how this qualitative information can be transformed into explicit quantitative results on solutions of these equations. The results are remotely resembling the "removable singularity" theorems from the classical complex variables theory.

Anna Zdunik: Random dynamics of polynomial and entire maps.

The study of random dynamics of holomorphic maps in the Riemann sphere was inspired by the seminal paper of E. Fornaess and N. Sibony (Random iterations of rational functions, Ergodic Theory and Dynamical Systems, 1991).

I will present some results about random dynamics of polynomials and entire maps. For example, I will consider random (or: non- autonomous) iteration of maps in quadratic family Q_c(z) = z^2 + c, and random iteration in the exponential family E(z) = exp(z).

In particular, the following questions will be addressed: connectedness of the Julia set, Hausdorff dimension of the Julia set and dimension of the radial Julia set and its dependence on the "range of randomness", dimension of the harmonic measure on the Julia set of random iteration of polynomials.

Andrew Zimmer: A metric analogue of Hartogs' theorem.

In this talk I will discuss a metric version of Hartogs' theorem where the holomorphic function is replaced by a locally symmetric metric. Then I will explain how this general result can be used to characterize the strongly pseudoconvex domains where the Kobayashi metric is a Kaehler metric.