Workshop on Dynamical systems in logic, complex analysis and ergodic theory.

Marco Abate: “Fatou flowers and parabolic curves”
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)

Jean Philippe Rolin: ” Fractal analysis and normal forms for power-log transseries”.

Abstract : “Classical methods allow, given a formal power series, to build a normal form and to embed it in the flow of a formal vector field. We show how these results can be extended to transseries whose monomials are products or real powers of the variable and its logarithm. We give some motivations arising from fractal analysis.”

Tamara Servi: “Oscillatory integrals and subanalytic geometry”
Abstract: We prove the stability under integration and under Fourier transform of a concrete class E of functions, containing all globally subanalytic functions and their complex exponentials. The class E is a system of algebras of which we describe explicitly the generators. The methods of proof pertain to subanalytic geometry (in particular, subanalytic resolution of singularities) and to the theory of almost periodic functions. This provides an example of a fruitful interaction between singularity theory, model theory and analysis.
Joint work with R. Cluckers, G. Comte, D. Miller and J.-P. Rolin.

Fabrizio Bianchi: “Holomorphic motions of Julia sets”

Abstract: Starting from the basics in holomorphic dynamics in one variable, I will review the classical theory by Mané-Sad-Sullivan about stability of rational maps and briefly present a generalization of this theory in higher dimension.

I will focus on the arguments that do not readily generalize in the second case, and introduce the tools and ideas that allow one to overcome these problems.

Miriam Benini “Repelling periodic points for transcendental entire functions”

Abstract: How many repelling periodic points of any period does a transcendental function have? For a generic rational function, the number of periodic points can be easily counted using the degree, and by the Fatou Shishikura inequality all but finitely many are repelling. Entire functions in general do not even need to have fixed points,see for example e^z+z. However we will be able to show-with a rather elementary proof- that for several important classes of transcendental functions there are-as expected-infinitely many repelling periodic points of any given period,and give some more information on the way they are distributed in the dynamical plane.

Alessandro Berarducci “Surreal numbers and H-fields”

Abstract: Given a field with a derivation, there is a notion of “differentiably closed field”, meaning, roughly, a field where one can solve as many differential equations as reasonably possible. We concentrate here on “H-fields”, which are ordered differential fields with certain compatibility properties between the order and the derivation. The main example are provided by: 1) Hardy fields; 2) o-minimal structures; 3) various notions of “transseries”, such as those studied by Ecalle to give a solution to the Dulac’s problem. The ground breaking work of van den Dries, Aschenbrenner and van der Hoeven (arXive 2015, 702 pages), provides the suitable axioms for a H-field with all the desired closure properties, and shows that the transseries are a model of the axioms. In a joint work with Vincenzo Mantova (arXive 2015) we endow Conway’s surreal numbers with a transserial structure and a compatible derivation, settling various previous conjectures both positively and negatively. Ecalle’s transseries embed in the surreal numbers as a differential field with an exponential map. Van den Dries, Aschenbrenner and van der Hoeven have recently proved that our derivation satisfies their axioms and makes Conway’s surreal numbers a universal domain for ordered H-fields. Joint work with Vincenzo Mantova.

Niels Langeveld “Ito continued fractions, Entropy and Matching”

Abstract: A lot of study has been done on the entropy of alpha continued fractions. Less study has been done on the entropy of Ito continued fractions.
Since these families of continued fraction maps are closely related, one might expect similar behavior for the entropy of the Ito continued fractions as for the alpha continued fractions. We will see similarities and point out differences. We end with simulations which will provide us some conjectures.
(Work in progress with C. Carminati)

Claudio Bonanno “The Farey map and its role in ergodic and spectral theory”

Abstract: I will discuss some results concerning the Farey map, and in particular the associated transfer operators. Main motivations come from the fact that the Farey map preserves an infinite invariant measure and from its applications to spectral theory of hyperbolic surfaces.

Han Peters “Henon maps with a dominated splitting”
Abstract: Consider the iteration of a polynomial without critical points on its Julia set. A result of Fatou states that all Fatou components of such a polynomial are eventually mapped onto a basin of a parabolic or attracting periodic cycle. In particular there are no wandering Fatou components. In joint work with Misha Lyubich we generalize this classical result to the setting of two-dimensional polynomial automorphisms. The condition on the critical points corresponds to the assumption that the map is substantially dissipative and admits a dominated splitting on its Julia set.

Carlo Carminati “Regularity and bifurcation phenomena in simple families of maps”

Abstract: We study the mechanism which leads to stability and bifurcation phenomena in a simple family of piecewise affine maps. This phenomenon is apparent from the self similar structure of the entropy function. In the case examined we detected also a phenomenon analogous to the tuning for the logistic family.

Stefano Galatolo “Rigorous approximation of the statistical properties of dynamics”
Abstract: Computers and computer aided proofs can help in answering to several mathematical questions on dynamics. If we are interested to topological or qualitative questions, many successful approaches are known. Much less is known about ergodic and “quantitative” statistical properties. We will show how the rigorous computation of invariant measures and other properties of the transfer operator can be approached. This helps to answer in suitable systems to questions regarding the statistical properties of dynamics.
In particular the following objects related to the statistical behavior of a system will be considered:
+ Lyapunov exponents (piecewise expanding and intermittent maps);
+ dimension of attractors (for some Lorenz like system);
+ speed of convergence to equilibrium (for systems satisfying a Lasota Yorke inequality);
+ linear response and diffusion coefficients (for systems satisfying a Lasota Yorke inequality).
In the seminar we will focus on some of them.
The result of some rigorous numerical experiment will also be shown.

Matteo Ruggiero Local dynamics of superattracting germs in dimension 2.

Abstract: The description of local dynamics for superattracting germs in dimension 1 is quite simple : up to change of coordinates, we can always reduce to the case of a power.

The study of normal forms in higher dimensions is much harder, mainly because of the geometrical degeneracy of the critical locus.

In a joint work with William Gignac, we present a quite precise description of the dynamics of superattracting germs in dimension 2.
This description is based on finding suitable birationally equivalent models of the plane, where the dynamics induced on exceptional primes is well behaved. To deal with such data, we study the dynamics induced on a suitable combinatorial object, the valuative tree.

Workshop on Dynamical systems in logic, complex analysis and ergodic theory.

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