Author: Yi Pan

Silvius Klein (Pontifical Catholic University of Rio de Janeiro)

In this talk I will discuss some recent results on large deviations type estimates for certain skew-product dynamical systems. Moreover, I will describe an abstract approach for deriving such statistical properties, based on the strong mixing properties of an appropriate Markov operator.[Based on joint work with Ao Cai (Soochow University) and Pedro Duarte (University of Lisbon).]

 Silvius Klein (Pontifical Catholic University of Rio de Janeiro)

In this talk I will discuss some recent results on large deviations type estimates for certain skew-product dynamical systems. Moreover, I will describe an abstract approach for deriving such statistical properties, based on the strong mixing properties of an appropriate Markov operator.
[Based on joint work with Ao Cai (Soochow University) and Pedro Duarte (University of Lisbon).]

Stefano Marmi  (Scuola Normale Superiore di Pisa)

Addressing a question raised by Kolmogorov and Herman, I will review previous work on the analytic structure in the complex frequency plane of solutions of certain small divisor problems and its relation with with the notion of quasianalytic function of a complex variable. I will discuss both the linear and nonlinear case.Most of the talk will be based on papers with C. Carminati, D. Sauzin and A. Sorrentino

Anna Florio (Université Paris Dauphine-PSL)

A celebrated result by Zehnder in the ’70s states that a generic analytic area-preserving map of the disk, having the origin as elliptic fixed point, exhibits a transverse homoclinc orbit in every neighborhood of the origin. In an ongoing project with Inmaculada Baldomà, Martin Leguil and Tere Seara, we adapt the strategy of Zehnder and use Aubry-Mather theory for twist maps in order to show that a generic analytic strongly convex billiard has, for every rational rotation number, a periodic orbit with a transverse homoclinic intersection.

Daniel Tsodikovich (ISTA)

We will discuss the work of Agapov, Bialy, and Mironov from 2016 about integrable geodesic flows on the torus, in which they prove two results: 

1. Any analytic Liouville metric without magnetic field can be deformed into a non-Liouville metric with non-zero magnetic field, the flow of which is integrable on one level set, with integral  quadratic polynomial in momentum variables.
2. Globally there is rigidity – if there is a quadratic polynomial integral for the whole torus for a magnetic geodesic flow, then up to some change of variables the flow has essentially a unique form.

Rafael Potrie (Centro de Matemática Universidad de la República, Uruguay)

I will try to give an overview of the problems in the understanding of the topological properties of partially hyperbolic diffeomorphisms in dimension 3 as well as the possible and known connections to their dynamics. I will concentrate on new examples and the challenges to understand their global dynamical properties. 

Susanna Terracini (University of Turin)

We deal, for the classical N-body problem, with the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We tackle the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs in a unified manner.  Our approach is based on the minimization of a renormalized Lagrangian action, on a suitable functional space. With this new strategy, we are able to confirm the already-known results of the existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a precise description of the growth of parabolic and hyperbolic-parabolic solutions. 

Finally, we will apply this novel variational approach to detecting oscillating solutions at large in the isosceles restricted three body problem. This is work in collaboration with Davide Polimeni and Jaime Paradela Diaz.

Speaker: Jianlu Zhang
Date: 15.07.2024 at 11:00

In 1987, Lions firstly proposed the homogenization for Hamilton-Jacobi equations, which revealed the significance of effective Hamiltonian in controlling the large time behavior of solutions. He also pointed out a vanishing discount procedure which is equivalent in obtaining the effective Hamiltonian, yet the convergence of solutions in this procedure was unknown until recently. In a bunch of joint works, we verified this convergence by using dynamical techniques.

Speaker: Qiaoling Wei
Date: 12.03.24 at 14:00

Abstract: Classically, for a local analytic diffeomorphism F of (R^2,0) with a non-resonant elliptic fixed point (eigenvalues exp(\pm2\pi i\omega) with \omega real irrational), one can find formal normalizations, i.e. formal conjugacies to a formal diffeomorphism invariant under the group of rotations. Less demanding is the notion of a “geometric normalization” that we introduce: this is a formal conjugacy to a formal diffeomorphism which maps any circle centered at 0 to a circle centered at 0. Geometric normalizations are not unique, but they correspond in a natural way to a unique formal invariant foliation. We then show that, generically, all geometric normalizations are divergent, so there is no analytic invariant foliation. The talk is based on joint works with Alain Chenciner, David Sauzin.