Second meeting PHC Project Galileo Transcendental dynamics from one to several complex variables

Liste des Contributions

1. Parabolic implosion in dimension 2 part 1

Matthieu Astorg (Université d'Orléans, IDP)

25/06/2025 09:00

Parabolic implosion is a tool for studying the dynamics of perturbations of a map with a fixed point tangent to the identity, or more generally with at least one eigenvalue which is a root of unity. We will start by surveying classical parabolic implosion in dimension one, and then we will explain an ongoing work on parabolic implosion of germs tangent to the identity in dimension 2.

Joint...

2. Dynamics of Fuchsian meromorphic connections with real periods

Marco Abate

25/06/2025 11:00

An interesting class of examples of holomorphic maps tangent to the identity in several complex variables is given by the time-1 maps of homogeneous
vector fields. It is known that the study of the dynamics of these maps can be reduced to the study of the dynamics of the geodesic field of meromorphic connections on Riemann surfaces. In this talk we shall describe some recent results, obtained...

3. On the dynamics of tangent-like mappings part 1

Núria Fagella

25/06/2025 14:00

In this talk we will introduce a transcendental version of the theory of polynomial-like mappings. The model family is a one parameter family $T_\alpha$ of "generalized tangent maps", which are meromorphic funtions with exactly two asymptotic values, only one of which is free. A straightenning theorem will explain why we find copies of Julia sets of $T_\alpha$ in the dynamical plane of other...

4. Hyperbolic components and iterated monodromy of polynomial skew-products of $\mathbb{C}^2$

Virgile Tapiero

25/06/2025 16:00

In this talk, we study the families $Sk(p, d)$ of polynomial skew-products $f(z, w) = (p(z), q(z, w))$ of degree $d > 1$, where $p(z)$ is fixed and $q(z, w)$ varies.
Astorg and Bianchi proved that the notion of hyperbolic components is meaningful in this setting, and they studied these components in detail for $d = 2$.

I will present my recent work on hyperbolic components in higher...

5. Free discussions

25/06/2025 17:00

6. Parabolic implosion in dimension 2 part 2

Matthieu Astorg (Université d'Orléans, IDP)

26/06/2025 09:00

Parabolic implosion is a tool for studying the dynamics of perturbations of a map with a fixed point tangent to the identity, or more generally with at least one eigenvalue which is a root of unity. We will start by surveying classical parabolic implosion in dimension one, and then we will explain an ongoing work on parabolic implosion of germs tangent to the identity in dimension 2.

Joint...

7. Equidistribution speed towards Green measures and currents

Marco Vergamini

26/06/2025 11:00

Let $f$ be a holomorphic endomorphism of a complex projective space. We study the action by pullback of the iterates of f on forms and currents. It is known from the work of Dinh-Sibony that they equidistribute towards the Green currents of f and that the speed of equidistribution is exponential when tested against Hölder continuous observables. Recently, as a key tool for the statistical...

8. On the dynamics of tangent-like mappings part 2

Núria Fagella

26/06/2025 14:00

In this talk we will introduce a transcendental version of the theory of polynomial-like mappings. The model family is a one parameter family $T_\alpha$ of "generalized tangent maps", which are meromorphic funtions with exactly two asymptotic values, only one of which is free. A straightenning theorem will explain why we find copies of Julia sets of $T_\alpha$ in the dynamical plane of other...

9. Bulging of wandering Fatou components for transcendental skew products

Tom Potthink

26/06/2025 16:00

The Fatou components of complex endomorphisms are subject of extensive study. Whereas the one-dimensional case admits a fairly complete classification, no such picture exists in higher dimensions. One approach towards better understanding the Fatou components of higher dimensional functions is to restrict oneself to a subclass of holomorphic functions, such as skew-products. While polynomial...

10. Free discussions

26/06/2025 17:00

11. Infinitely many ergodic measures with infinite entropy

Anna Miriam Benini

27/06/2025 09:30

Let $f$ be a transcendental entire function in class $B$ with finite order of growth.
We show that we can construct infinitely many ergodic measures with infinite entropy whose support is the Julia set. This is in contrast with the rational setting in which the measure of maximal entropy is unique. This is joint work with Leandro Arosio, John Erik Fornaess and Han Peters.

12. Horn maps of semi-parabolic Hénon maps

Fabrizio Bianchi

27/06/2025 11:00

We prove that horn maps associated to quadratic semi-parabolic fixed points of Hénon maps, first introduced by Bedford, Smillie, and Ueda, satisfy a weak form of the Ahlfors island property. As a consequence, two natural definitions of their Julia set (the non-normality locus of the family of iterates and the closure of the set of the repelling periodic points) coincide. As another...