Author: Yi Pan

Marco Mazzucchelli (ENS Lyon)

In this talk, based on joint work with Marcelo Alves, I will present three new theorems on the dynamics of geodesic flows of closed Riemannian surfaces, proved using the curve shortening flow. The first result is the stability, under C^0-small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for closed geodesics on orientable closed Riemannian surfaces. The third theorem asserts the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface.

Umberto Leone Hryniewicz (RWTH Aachen University)

We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many periodic orbits, if the first Chern class of the associated contact structure is torsion. This result covers Reeb flows on the three-sphere, and implies that every Finsler metric on a closed surface has either two or infinitely many closed geodesics. Joint work with Cristofaro-Gardiner, Hutchings and Liu.

Miguel Garrido (Universitat Autònoma de Barcelona)

Consider a one-parameter family of smooth surface diffeomorphisms unfolding a quadratic homoclinic tangency to a hyperbolic fixed point. It is well known that in this unfolding there exists a Newhouse domain, i.e. an open set of parameters for which the corresponding diffeomorphisms exhibit persistent homoclinic tangencies.
The analysis of the unfolding is more subtle when the homoclinic tangency is associated to a persistent degenerate saddle, i.e. a parabolic fixed point which exists for all values of the parameter and for which the topological picture of the local dynamics resembles that of a hyperbolic fixed point (existence of invariant manifolds and C^0 Lambda lemma).
These degenerate saddles appear naturally in several models in Celestial Mechanics, in particular, in the so-called Restricted 4-Body Problem. We prove that, in a particular configuration of the latter model which can be reduced to an area preserving map, there exists a degenerate saddle with a quadratic homoclinic tangency that unfolds generically (as we move the masses of the bodies). Moreover, and despite the fact that the C^1 Lambda lemma does not hold for this degenerate saddle, we show that the dynamics at the unfolding of the tangency can be renormalized, with the critical Hénon map showing up in the limit process.
This implies the existence of a Newhouse domain in the parameter space (the masses of the bodies) and a residual subset of parameters for which there exist hyperbolic sets of large Hausdorff dimension which are accumulated by elliptic islands.
This is joint work with Pau Martín and Jaime Paradela.

Benoît Grébert (Université de Nantes)

I will present an almost global existence result for semilinear Hamiltonian PDEs on compact boundaryless manifolds. As a main application, we prove the almost global existence of small solutions to nonlinear Klein–Gordon equations on such manifolds: for almost all mass, any arbitrarily large r and sufficiently large s, solutions with initial data of sufficiently small size ε ≪ 1 in the Sobolev space Hs ×Hs−1 exist and remain in Hs ×Hs−1 for polynomial times |t| ≤ ε−r. Surprisingly, it turns out that the geometry of the manifold has no influence. The abstract result applies to equations satisfying very weak non resonance conditions and natural multilinear estimates. (Joint work with D. Bambusi, J. Bernier and R. Imekraz)

Jacopo De Simoi (University of Toronto)

Given a planar domain with sufficiently regular boundary, one
can study periodic orbits of the associated billiard problem. Periodic
orbits possess a rich and intricate structure: it is then natural to ask
how much information about the domain is encoded in the set of lengths
of such orbits. The quantum analog of this question is the celebrated
Laplace inverse problem, or “Can one hear the shape of a drum?”

We prove Marked Dynamical Spectral Determination among
ℤ₂-symmetric smooth convex domains close to the disk: if any two such
domains have the same Marked Length Spectrum, they must necessarily be
isometric domains.  This substantially improves the deformational result
obtained in a prior work with Kaloshin and Wei.

Dario Bambusi (Università degli studi di Milano)

It is well known that solutions of the Nonlinear Klein Gordon equation $$\frac{1}{c^2}u_{tt}-u_{xx}+c^2 u\pm u^3=0\ ,\quad u(0,t)=u(\pi,t)=0$$ are well approximated in the non relatistic limit ($c\to\infty$)by solutions of the cubic Nonlinear Schr\”odinger equation.
In the present talk I will present a result based on KAM theory, according to which the quasiperiodic solutions of NLKG constructed through KAM theory converge {\it uniformly for $t\in\R$} to solutions of NLS.
I will start the presentation by recalling some classical results on the justification of the NLS as the classical limit of NLKG, then I will give a precise statement of the result and the main ideas of the proof. 
Joint work with Andrea Belloni and Filippo Giuliani.

Mauricio Poletti (Universidade Federal do Ceará)

For $C^(l+)$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. In this talk I will give two the applications: Uniqueness of MME for finite horizon dispersing billiards and the robustly non-uniformly hyperbolic volume-preserving endomorphisms introduced by Andersson-Carrasco-Saghin.

Yunzhe Li (ISTA)

We construct a nontrivial analytic family of standard maps in which the Lyapunov exponents of infinitely many periodic orbits remain constant throughout the family. After a brief review of the background and related results, I will discuss key components of the proof, including a resonant normal form for discrete maps and an iterated correction mechanism.

Sébastien Biebler (Université Paris Cité)

One of the main goals in the theory of dynamical systems is to describe the dynamics of a “typical” map. For instance, in the case of diffeomorphisms of a given manifold, it was conjectured by Smale in the 60s that uniform hyperbolicity was generically satisfied. This hope was however fast discouraged by exhibiting dynamical systems displaying in a robust way dynamical configurations which are obstructions to hyperbolicity: robust homoclinic tangencies (this is the so-called Newhouse phenomenon) and robust heterodimensional cycles. In this talk, I will explain these phenomena and their extensions to the complex setting. In particular, I will show how to construct robust heterodimensional cycles in the family of polynomial automorphisms of C³. The main tool is the notion of blender coming from real dynamics.

Jérôme Buzzi (CNRS, Laboratoire de Mathématiques d’Orsay)

Santiago Martinchinch introduced Discretized Anosov flows, a large class of partially hyperbolic diffeomorphisms that includes deformations of time-one maps of transitive Anosov flows. Under an assumption of irreducibility, we establish the following dichotomy

– either there are exactly two ergodic measures maximizing the entropy (or MMEs);
 – or there is a unique MME with zero center exponent.
The hyperbolic case is open and dense and implies that the MMEs are Bernoulli and exponentially mixing (they satisfy Strong Positive Recurrence).
This is a joint work in progress with Sylvain CROVISIER, Mauricio POLETTI, and Ali TAHZIBI.