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.”
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”
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)
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.
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.
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.