Author: Mathieu Helfter

Martin Leguil (CMLS, Polytechnique)

Abstract: When studying dynamical systems of geometric origin, a fundamental question is whether periodic data—particularly the length spectrum—encodes the underlying geometry of the system. In this talk, I will present several results addressing this question for billiards with hyperbolic behavior. To give more structure to the length spectrum, it is common to augment it with symbolic data, resulting in a marked length spectrum (MLS). I will discuss the MLS-rigidity for two classes of domains: open dispersing billiards satisfying the no-eclipse condition, and Sinai billiards on the two-torus. While the first class exhibits nice symbolic coding, interesting dynamics is confined to a Cantor set, and then analyticity is required to recover the geometry from the MLS. On the contrary, for Sinai billiards with finite horizon, periodic orbits are dense in phase space, but the lack of structural stability complicates the marking of the spectrum. I will describe a CAT(0)-based approach to the MLS-rigidity of Sinai billiards, leveraging the approximation of the billiard flow by Anosov geodesic flows. Joint work with De Simoi-Kaloshin, and Finamore.

Pau Martín (UPC)

Abstract: In this work, we prove that a generic unfolding of an analytic Hamiltonian Hopf singularity (in an open set with codimension 1 boundary) possesses transverse homoclinic orbits for subcritical values of the parameter close to the bifurcation parameter. As a consequence, these systems display chaotic dynamics with arbitrarily large topological entropy. We verify that the Hamiltonian of the restricted planar circular three-body problem (RPC3BP) close to the Lagrangian point $L_4$ falls within this open set. The generic condition ensuring the presence of transversal homoclinic intersections is subtle and involves the so-called Stokes constant. Thus, in the case of the RPC3BP close to $L_4$, our result holds conditionally on the value of this constant.

Rafael Ramírez-Ros (Universitat Politècnica de Catalunya, Spain)

We find necessary and sufficient conditions for high-order persistence of
resonant caustics in perturbed circular billiards. The main technical tool
is a perturbation theory based on the Bialy-Mironov generating function
for convex billiards. A caustic is a curve such that any billiard
trajectory, once tangent to the curve, stays tangent after every reflection. A convex
caustic is p/q-resonant when all its tangent trajectories form closed
polygons with q sides that make p turns around the caustic. We prove
that any resonant caustic of the circular billiard with period q persist up to order
⌈q/n⌉ − 1 under any ‘polynomial’ perturbation of the circle of degree.
(This is a join work with Comlan Edmond Koudjinan.)