I will discuss the role of classical results about homoclinic tangles in the question of existence and multiplicity of Gibbs measures on certain systems.

In this talk, we consider a new stability result for the topological entropy of Hamiltonian diffeomorphisms on closed surfaces. Topological entropy is a fundamental measure of orbital complexity in dynamical systems, capturing chaotic behavior through a single non-negative value. Based on Floer-theoretic persistence, we discuss the persistence of this invariant. As a corollary, we show that if a Hamiltonian diffeomorphism has positive entropy, then any perturbation supported in a disk of sufficiently small area still has positive entropy. This talk is based on joint work with Beomjun Sohn and Matthias Meiwes.

In this talk, we present rigidity properties of autonomous Hamiltonian systems which are Zoll (that is, they induce a free circle action) on a sequence of energies converging to a symplectic Morse-Bott minimum. We then specialize these techniques to magnetic systems on closed manifolds with symplectic magnetic form. In this setting, we show that the magnetic system defines an almost Kähler structure, whose holomorphic sectional curvature, corrected by a term measuring the non-integrability of the almost complex structure, is constant. In particular, we obtain a dynamical characterization of complex space forms among Kähler manifolds. This is joint work with Johanna Bimmermann and Samanyu Sanjay.

Using generating functions quadratic at infinity for Lagrangian submanifolds of twisted cotangent bundles, we define spectral selectors for compactly supported lcs Hamiltonian diffeomorphisms of the locally conformal symplectizations \(M\) and \(M'\) of the standard contact manifolds \(\mathbb R^{2n+1}\) and \(\mathbb R^{2n} \times S^1\) respectively, and obtain several applications : the construction of a bi-invariant partial order on the group of compactly supported lcs Hamiltonian diffeomorphisms of \(M\) and \(M'\), of an integer-valued bi-invariant metric on the group of compactly supported lcs Hamiltonian diffeomorphisms of \(M'\), and of an integer-valued lcs capacity for domains of \(M'\). The lcs capacity is used to prove a lcs non-squeezing theorem in \(M'\) analogous to the contact non-squeezing theorem in \({\mathbb R}^{2n} \times S^1\) discovered in 2006 by Eliashberg, Kim and Polterovich.

Given a Morse-Smale pair on a manifold , it is possible to entirely recover its fundamental group in a combinatorial manner. We call this construction the Morse fundamental group. Motivated by a similar construction of a "Floer fundamental group" by Barraud, and by the many uses of continuation maps in symplectic topology, I will explain in this talk how continuation maps give us functoriality and invariance of the Morse fundamental group, and what the differences are with the usual homological setup. I will then, if time permits, talk about how this construction can be applied to the study of Legendrians thanks to the Morse theory of their generating functions, as well as some dynamical implications.

Consider a displaceable Lagrangian submanifold. In particular, its displacement energy is finite. We study how the displacement energy changes under stabilization, that is, taking product with a non-displaceable Lagrangian, eg., the \(0\)-section in \(T^*S^1\). We present criteria to classify Lagrangians whose displacement energy does not change under stabilization in terms of Floer theoretic invariants. This allows us to fully classify Lagrangians which exhibit this property in certain toric manifolds. This is based on ongoing work with Jean-Philippe Chassé (CRM Montreal).

We consider a functional \(E \geq 0\) (originally studied by Chern and Hamilton) on the space of metrics compatible with a stable Hamiltonian structure (SHS). In dimension 3 we show that if a metric is critical for this functional, either \(E=0\) and the SHS is a normal almost contact metric structure, or \(E>0\) and the Reeb flow is algebraic Anosov and the SHS is an invariant structure on the corresponding Lie group. This unifies the understanding in the contact case (Mitsumatsu, Peralta-Salas, Slobodeanu and Hozoori) and the cosymplectic case (D., González-Prieto, Miranda, Peralta-Salas).

We study the three-body problem with masses \(m_0=1\), \(m_1=\mu\) and \(m_2=\mu\), with \(0<\mu <<1.\) We show the existence of a one-parameter family of quasiperiodic solutions, parametrized by the angular momentum \(c\). This family of quasiperiodic solution starts, at \(c=0\) from a genuine collinear periodic solution with two binary collisions at each period, that is known as Von Schubart's orbit. When \(c>0\), the corresponding solution of the family is obtained as a continuation with respect to the parameter \(\mu\) of a periodic solution of two uncoupled Kepler problem, where the two small bodies are located on two Keplerian ellipses, symmetric with respect to the origin, where at time \(t=0\) the first body is at the perihelion of the first ellipse, and the second body is at the aphelion of the second ellipse. This family is obtained by using the classical continuation method, and each solution is actually periodic in a moving frame that is rotating with small angular velocity. This is a joint work with Anete Soarse Cavalcanti.

My research is motivated by the following practical question: can a rocket travel between any two points in the gravitational field of the Earth of the Moon, using its engines only at the beginning and at the end of the journey? This question, known as the two-boost problem, arises naturally in the context of space mission design. In this talk I will show how to answer this question using the algebraic techniques of Lagrangian Rabinowitz Floer homology.