Author: Daniel Tsodikovich

Michael Entov (Technion – Israel Institue of Technology)

Abstract: I will discuss how a filtered version of the Legendrian contact homology (a major object of study in contact topology) can be used to obtain new information about contact Hamiltonian flows on contact manifolds. The applications concern conformal factors of contactomorphisms and trajectories of contact Hamiltonian flows connecting two disjoint Legendrian submanifolds or starting on one of them and asymptotic to the other.

This is a joint work with L.Polterovich.

Eran Igra, Valerii Sopin (SIMIS – Shanghai Institute for Mathematics and Interdisciplinary Sciences)

Consider a rectangle map $H:ABCD\to\mathbb{R}^2}$ s.t. $H$ is a Smale Horseshoe map. Now, consider another map $G:ABCD\to\mathbb{R}^2$ s.t. $G$ is the identity (or more generally, some other homeomorphism with “simple” dynamics). By a classical result due to James Yorke and Kathleen Alligood we know that as we deform $G$ isotopically to $H$, more and more periodic orbits appear via period-doubling cascades, culminating in the chaotic dynamics of the Smale Horseshoe – i.e., we have a period-doubling route to chaos. That being said, due to the two-dimensional structure of the phase space we are free to choose many isotopies deforming $G$ to $H$ – and all feature a period-doubling route to chaos. This leads us to ask the following, natural question: can we somehow differentiate between different two-dimensional period-doubling routes to chaos?

It is precisely this question that we study in this talk. Using a mixture of Representation-Theoretic, Complex-Analytic, and Braid Theoretical tools we define topological invariants that describe how a given isotopy progresses from order into chaos. Following that, we exemplify how these invariants can be used to analyze the dynamical complexity of certain examples, including the Henon map – if time permits, we also discuss how these ideas can be generalized to a wider context.

Based on work in progress.

Dongchen Li (Imperial College London)

A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some central subspace has a non-empty interior and thus has a higher topological dimension than the set itself.We show that, for any symplectic C^r-diffeomorphism (where r is sufficiently large and finite, or r=∞,ω) of a 2N-dimensional (N>1) symplectic manifold, symplectic blenders can be obtained by an arbitrarily small symplectic perturbation near any one-dimensional whiskered KAM-torus that has a homoclinic orbit. Using this result, we prove that non-transverse homoclinic intersections between invariant manifolds of a saddle-center periodic point (i.e., it has exactly one pair of complex multipliers on the unit circle) are persistent in the following sense: the original map is in the C^r closure of a C¹ open set in the space of symplectic C^r-diffeomorphisms, where maps having such saddle-center homoclinic intersections are dense. These results also hold for Hamiltonian flows in the corresponding settings.

Gil Bor (CIMAT)

This talk is concerned with a billiard version of Jacobi’s Last Geometric Statement and its generalizations. Given a point O inside an oval billiard table (or mirror), one considers the family of rays emanating from O and the caustic (or envelope) of the reflected family of rays after n reflections off the walls of the table. I will describe two related statements:

(1) Theorem: for a generic O this caustic has at least 4 cusps for each positive integer n. 

(2)  Conjecture: for an elliptic table there are exactly four (ordinary) cusps. 

I will describe a proof of (1) and  partial results concerning  (2). 

This is joint work with Mark Spivakovsky (Toulouse) and Serge Tabachnikov (Penn State). 

References:

* https://arxiv.org/abs/2112.07852

* https://arxiv.org/abs/2406.11074.

Adi Dickstein (Tel Aviv University)

Topological quasi-states are special functionals on the algebra of continuous functions which are linear on single-generated subalgebras. They trace their origins to the von Neumann axioms of quantum mechanics. On symplectic surfaces, every topological quasi-state is symplectic, i.e., linear on Poisson-commutative subalgebras. We discuss the failure of this phenomenon in higher dimensions based on the study of symplectic embeddings of polydiscs. Furthermore, we introduce a Wasserstein-type metric on quasi-states and use it for quantitative constraints on symplectic quasi-states. The talk is based on a joint work with Frol Zapolsky.

Corentin Fierobe (University of Rome Tor Vergata)

Mathematical billiards in strictly convex domains with smooth boundaries provide tangible examples of twist maps on the cylinder, where the dynamics exhibit “almost integrable” behavior near the boundary. Building on this idea, Lazutkin demonstrated the existence of a Cantor set of positive measure, which contains zero, and within which the billiard maps have invariant curves associated with certain rotation numbers. These invariant curves evolve smoothly as the rotation number varies, in the Whitney sense. In this talk, I will present a generalization of this result for billiards with analytic boundaries, a joint work with Frank Trujillo and Vadim Kaloshin, motivated by recent advances from Carminati, Marmi, Sauzin, and Sorrentino. This extension shows that the Cantor set of rotation numbers can be continued into the complex plane, with the complex counterpart containing structures known as “diamonds.” This discovery offers new insights into length spectral rigidity.

Olga Bernardi (University of Padova)

Symplectic billiards were introduced by P. Albers and S. Tabachnikov in 2018 as a simple dynamical system where, opposed to Birkhoff billiards, the generating function is the area instead of the length. We first recall the main properties of symplectic billiards in strictly-convex domains. We proceed by presenting 
this recent result: if the phase-space is fully foliated by continuous invariant curves which are not null-homotopic, then the boundary of the billiard table is an ellipse.
 We finally discuss the state of the art 
both of integrability and of area spectral rigidity for these billiards. 
Joint works with Luca Baracco and Alessandra Nardi.