13 January, 2023 (16:00 GMT)

Alexander Grigo (University of Oklahoma)

"Existence of stable orbits in smooth stadium billiards"

In 1973 Lazutkin showed that for strictly convex billiard tables with smooth enough boundary there exists an uncountable family of caustics near the boundary. The presence of these caustics prevents the billiard dynamics from being ergodic and gives rise to nearly integrable motion close to the boundary.

Shortly after, in 1974 Bunimovich showed that there are convex billiard tables on which the billiard dynamics is hyperbolic and ergodic. In billiards with focusing boundary components the mechanism that creates the hyperbolicity is the mechanism of defocusing. The most famous and best studied convex billiard in this class is the stadium.

These two results are in sharp contrast to each other, and a natural question that arose almost immediately is: Which degree of smoothness of the boundary of the billiard table separates convex billiards with completely chaotic dynamics from non-ergodic dynamics with elliptic islands?

In this talk I present joint work with L.Bunimovich where we put forward a solution to this long standing problem. We show a number of results on the existence of stable periodic orbits as soon as the boundary of the billiard table is of class C^2. A key observation behind our results is the fact that as soon as the smoothness of the boundary is C^2 the resulting focusing boundary components fail to be absolutely focusing. The latter is the key ingredient in the standard construction of chaotic billiards with focusing boundary components.

(This is joint work with L.Bunimovich)

10 February, 2023 (13:00 GMT)

Irene De Blasi (University of Turin)

"Refraction galactic billiards: stability, rotation numbers and chaos"

A particular class of billiards, designated as galactic, is taken into consideration. 

Given a bounded domain D, suppose that a harmonic oscillator-type potential acts outside it, while a central point-mass generates a Keplerian potential in its interior. This kind of model can describe the motion of a particle in an elliptic galaxy having a central mass, such as a Black Hole, in its center. 

The associated trajectories are concatenations of Keplerian hyperbolæ and harmonic elliptic arcs, connected by Snell’s refraction rule on the boundary; the features of this dynamics, including its good definition, depend crucially on the boundary of D, which in general could be any C1 curve. 

During the seminar I will present results on the stability of the equilibrium trajectories for general domains and on the orbits’ rotation numbers for nearly-circular ones; in particular, these last outcome is obtained by applying results coming from KAM and Aubry-Mather theories. 

I will conclude by showing that, under very general conditions on D, the (topological) chaoticity of the associated billiard motion is ensured; this result is based on the construction of a suitable symbolic dynamics, that conjugates the billiard map to the Bernoulli shift (Devaney, 1994). 

This is a joint work with V. L. Barutello and S. Terracini. 

References:

Refraction Periodic Trajectories in Central Mass Galaxies, I. D.B., S. Terracini, Nonlinear Analysis 

On some Refraction Billiards, I. D.B., S. Terracini, Discrete and Continuous Dynamical Systems 

Chaotic dynamics in galactic refraction billiards, V. Barutello, I. D.B., S. Terracini, Preprint

10 March, 2023 (13:00 GMT)

Misha Bialy (Tel Aviv University)

"On Birkhoff conjecture for convex billiards and beyond"

I shall discuss recent progress in Birkhoff conjecture for convex centrally symmetric billiards.

Our approach is based on the so called Hopf rigidity discovered by E. Hopf for Riemannian Tori

without conjugate points. I shall explain why and how this phenomenon can be adapted to convex billiards. Our results also imply the rigidity of Mather beta-function of ellipse.

Moreover, this approach can be made effective, i.e. leading to sharp estimates on the measure of the invariant set occupied by minimal orbits.

This talk is based on several papers joint with Andrey Mironov and Daniel Tsodikovich.

28 April, 2023 (13:00 GMT)

Alexey Glutsyuk (Ecole Normale Supérieure de Lyon)

"On rationally integrable planar dual and projective billiards"

A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. 

The famous Birkhoff Conjecture  states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then the billiard is an ellipse. It was studied by many mathematicians, including H.Poritsky, M.Bialy, S.Bolotin, A.Mironov, V.Kaloshin, A.Sorrentino and others. 

 We study its following generalized dual version  stated by S.Tabachnikov. Consider a closed smooth strictly convex curve $\gamma\subset\mathbb{RP}^2$ equipped with a dual billiard structure: a family of non-trivial projective involutions acting on its projective tangent lines and fixing the tangency points. 

Suppose that its outer neighborhood admits a foliation by closed curves (including $\gamma$) such that the involution of each tangent line permutes its intersection points with every leaf. Then $\gamma$ and the leaves are conics forming a pencil.

 We prove positive answer in the case, when the curve $\gamma$ is $C^4$-smooth and the foliation  admits a rational first integral. To this end, we show that each 

 $C^4$-smooth germ $\gamma$ of planar curve carrying a rationally integrable dual  

 billiard structure (i.e., with involutions preserving  restrictions  to lines of some rational function) is a conic and classify all the rationally integrable dual  billiards on conics. 

 They include  the dual  billiards  induced by pencils of conics,  two infinite series of exotic dual billiards and five more  exotic ones.

If the time allows, we will briefly discuss recent result on classification of rationally integrable dual multibilliards (collections of some dual billiards and some points equipped with appropriately defined dual billiard structures at them). And the dual result: classification of piecewise smooth projective billiards with billiard flow admitting a first integral that is a rational 0-homogeneous function of the velocity.

12 May, 2023 (13:00 GMT)

Claudio Bonanno (Università di Pisa )

“The billiard-like motion of waves in stratified fluids”

We consider the motion of gravity waves generated by the perturbation of an incompressible fluid in a two-dimensional domain for fluids which are stably stratified with a variation of density in the direction of gravity. The dynamics of these waves may be described in terms of the motion in the domain of a point particle with non-standard reflection laws. In the talk I will describe the connections of the problem with the theory of circle homeomorphisms with break points and show the results in full details for the case of a fluid in a trapezoidal domain. Based on joint work with Giampaolo Cristadoro and Marco Lenci.

9 June, 2023 (13:00 GMT)

Illya Koval (IST Austria)

"Local strong Birkhoff conjecture of almost every ellipse"

The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. We will consider a stronger notion of integrability, namely, integrability close to the boundary, and explain the proof of a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse.

11 November, 2022 (13:00 GMT)

Sergei Tabachnikov (Penn State University) 

“Billiard-like maps and Dowker-style theorems"

The classic Dowker theorem concerns the minimal areas A(n) of n-gons circumscribed about an oval. The statement is that this sequence is convex:  A(n-1)+A(n+1) > 2A(n). These polygons are n-periodic orbits of the outer billiard about the oval. I shall show that this, and a number of similar geometric inequalities, are consequences of Mather's inequality for the minimal action of monotone twist maps. In particular, the four Dowker-style inequalities for inscribed/circumscribed polygons of maximal/minimal areas/perimeters are related to four billiard-like maps, three of which have been studied to variable degree, and one -- outer billiard with the length as the generating function -- was not studied yet. I shall mention other applications: to magnetic billiards, wire billiards, etc.

2 December, 2022 (14:00 GMT)

Leonid Bunimovich (Georgia Tech)

"Elliptic Flowers Billiards"

A new class of billiards, which proved to have a new type of never seen before dynamics will be introduced . I will also formulate several problems, some new, some a kind of traditional ones,  which arise (as always when we see systems with a new type of dynamics).