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Billiards in polygons are related to dynamics of the linear flow on flat surfaces. Through some examples of counting problems on flat surfaces and on moduli spaces of flat surfaces, we will see how combinatorics can lead to interesting dynamical results in this setting.

30F30 ; 32G15 ; 37D50

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Tiling billiards is a dynamical system where beams of light refract through planar tilings. It turns out that, for a regular tiling of the plane by congruent triangles, the light trajectories can be described by interval exchange transformations. I will explain this surprising correspondence, give related results, and show computer simulations of the system.

37D50 ; 37B50

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In this talk we explain how the Fibonacci trace map arises from the Fibonacci substitution and leads to a unified framework in which a variety of models can be studied. We discuss the associated foliations, hyperbolic sets, stable and unstable manifolds, and how the intersections of the stable manifolds with the model-dependent curve of initial conditions allow one to translate dynamical into spectral results.

81Q10 ; 81Q35 ; 37D20 ; 37D50

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We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Then we discuss applications such as proving various mixing properties of infinite measure preserving systems. Based on joint work with Dmitry Dolgopyat.[-]

We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Then we discuss applications such as proving various mixing properties of infinite measure preserving systems. Based on joint work with Dmitry ...[+]

37A50 ; 37D50 ; 60F05 ; 37D20

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We will discuss an approach to the statistical properties of two-dimensional dispersive billiards (mostly discrete-time) using transfer operators acting on anisotropic Banach spaces of distributions. The focus of this part will be our recent work with Mark Demers on the measure of maximal entropy but we will also survey previous results by Demers, Zhang, Liverani, etc on the SRB measure.

37D50 ; 37C30 ; 37B40

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In an ordinary billiard trajectories of a Hamiltonian system are reflected after a collision with a hypersurface (scatterer). If the scatterer has codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear e.g. as limits of systems with singularities in celestial mechanics. We discuss the existence of trajectories of such systems shadowing trajectories of a degenerate billiard.

37D05 ; 37D50 ; 37J10 ; 70F16

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Periodic trajectories in polygons are rare and beautiful, and have several surprising properties. During the lockdown period in 2020, we wrote a Sage program to draw every periodic trajectory on every regular polygon with an odd number of edges. I'll explain the underlying structure, and show lots of pictures from this vast menagerie. This is joint work with Samuel Lelièvre.

37E35 ; 37E05 ; 37D50

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Can a billiard map be locally conjugated to a rigid rotation? We prove that the answer to this question is positive in the category of formal series. We also present numerical evidence that for "good" rotation angles the answer is also positive in an analytic category.
billiard systems # integrable Hamiltonian systems # normal form convergence # small divisors # elliptic fixed point # analytic conjugacy

37D50 ; 70H06

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We will present a geometric criterion for the ergodicity of the billiard flow in a polygon with non-rational angles and discuss its application to the Diophantine case.

37D40 ; 37D50 ; 30F10 ; 30F60 ; 32G15

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We investigate the diffusion and statistical properties of Lorentz gas with cusps at flat points. This is a modification of dispersing billiards with cusps. The decay rates are proven to depend on the degree of the flat points, which varies from $n^{-a}$, for $ a\in (0,\infty)$. The stochastic processes driven by these systems enjoy stable law and have super-diffusion driven by Lévy process. This is a joint work with Paul Jung and Françoise Pène.[-]

We investigate the diffusion and statistical properties of Lorentz gas with cusps at flat points. This is a modification of dispersing billiards with cusps. The decay rates are proven to depend on the degree of the flat points, which varies from $n^{-a}$, for $ a\in (0,\infty)$. The stochastic processes driven by these systems enjoy stable law and have super-diffusion driven by Lévy process. This is a joint work with Paul Jung and Françoise ...[+]

37D50 ; 37A25 ; 60F05 ; 37D25

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Documents 37D50 13 résultats

Flat surfaces and combinatorics - Goujard, Élise (Auteur de la conférence) | CIRM H Nouveau

Interval exchange transformations from tiling billiards - Davis, Diana (Auteur de la conférence) | CIRM H Nouveau

A perspective on the The Fibonacci trace map - Damanik, David (Auteur de la conférence) | CIRM H Nouveau

​​​​Mixing and the local central limit theorem for hyperbolic dynamical systems - Nándori, Péter (Auteur de la conférence) | CIRM H Nouveau

Transfer operators for Sinai billiards - lecture 3 - Baladi, Viviane (Auteur de la conférence) | CIRM H Nouveau

Degenerate billiards - Bolotin, Sergei (Auteur de la conférence) | CIRM H Nouveau

Billiards on regular polygons - Davis, Diana (Auteur de la conférence) | CIRM H Nouveau

Billiard and rigid rotation - Treschev, Dmitry (Auteur de la conférence) | CIRM H Nouveau

On the ergodicity of billiards in non-rational polygons - Forni, Giovanni (Auteur de la conférence) | CIRM H Nouveau

​Levy diffusion of dispersing billiards with flat points - Zhang, Hong-Kun (Auteur de la conférence) | CIRM H Nouveau

Mixing and the local central limit theorem for hyperbolic dynamical systems - Nándori, Péter (Auteur de la conférence) | CIRM H Nouveau

Levy diffusion of dispersing billiards with flat points - Zhang, Hong-Kun (Auteur de la conférence) | CIRM H Nouveau