Degenerate billiards in celestial mechanics

In an ordinary billiard trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. This research is motivated by the problem of second species solutions of Poincaré.     

Periodic trajectories in 3-dimensional convex billiards

 We give a lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. Convex plane billiards were studied by G. Birkhoff, and the case of higher dimensional billiards is considered in our previous papers. We apply a topological approach based on the calculation of cohomology of certain configuration spaces of points on 2-sphere. Received: 11 June 2001 / Revised version: 26 February 2002     

Geometric-Dynamic Approach to Billiard Systems: I. Projective Involution of a Billiard,Direct and Inverse Problems

We suggest a geometric-dynamic approach to billiards as a special kind of reversible dynamic system and establish their relation to projective transformations (involutions) in the framework of this approach. We state the direct and inverse problems for billiards and derive equations determining the solutions of these problems in general form. Some simplest billiard involutions are calculated. We establish functional relations between the involution of a billiard, the equation for its boundary, and the field of normals to the boundary. We show how the involution is related to the curvature of the billiard boundary.     

Estimates for correlations in billiards with large arcs

Bunimovich billiards are ergodic and mixing. However, if the billiard table contains very large arcs on its boundary then if there exist trajectories experience infinitely many collisions in the vicinity of periodic trajectories on the large arc. The hyperbolicity is nonuniform and the mixing rate is very slow. The corresponding dynamics are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. The study of mixing rates in intermittent chaotic systems is mo...     

Topological Obstacles to the Realizability of Integrable Hamiltonian Systems by Billiards

Doklady Mathematics - We introduce the following classes of integrable billiards: elementary billiards, topological billiards, billiard books, billiards with a potential, with a magnetic field, and...     

Billiards in rational polygons: Periodic trajectories, symmetries, and d-stability

Periodic trajectories of billiards in rational polygons satisfying the Veech alternative, in particular, in right triangles with an acute angle of the form π/n with integern are considered. The properties under investigation include: symmetry of periodic trajectories, asymptotics of the number of trajectories whose length does not exceed a certain value, stability of periodic billiard trajectories under small deformations of the polygon. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 66–75, July, 1997. Translated by V. N. Dubrovsky     

Polygons in billiard orbits

We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these polygons.     

Topological Modeling of Integrable Systems by Billiards: Realization of Numerical Invariants

Doklady Mathematics - A local version of A.T. Fomenko’s conjecture on modeling of integrable systems by billiards is formulated. It is proved that billiard systems realize arbitrary numerical...     

Artin Billiard: Exponential Decay of Correlation Functions

The hyperbolic Anosov C-systems have an exponential instability of their trajectories and as such represent the most natural chaotic dynamical systems. The C-systems defined on compact surfaces of the Lobachevsky plane of constant negative curvature are especially interesting. An example of such a system was introduced in a brilliant article published in 1924 by the mathematician Emil Artin. The dynamical system is defined on the fundamental region of the Lobachevsky plane, which is obtained by identifying points congruent with respect to the modular group, the discrete subgroup of the Lobachevsky plane isometries. The fundamental region in this case is a hyperbolic triangle. The geodesic trajectories of the non-Euclidean billiard are bounded to propagate on the fundamental hyperbolic triangle. Here, we present Artin’s results, calculate the correlation functions/observables defined on the phase space of the Artin billiard, and show that the correlation functions decay exponentially with time. We use the Artin symbolic dynamics, differential geometry, and the group theory methods of Gelfand and Fomin.     

Classical Integrable Systems and Billiards Related to Generalized Jacobians

We study some classical integrable systems of dynamics (the Euler top in space, the asymptotic geodesic motion on an ellipsoid) which are linearized on unramified coverings of generalized Jacobian varieties. We find explicit expressions for so called root functions living on such coverings which enable us to solve the problems in terms of generalized theta-functions. In addition, general and asymptotic solutions for ellipsoidal billiards and the billiard in an ellipsoidal layer are obtained.     

On the model of non-holonomic billiard

In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.     

An Expansion Estimate for Dispersing Planar Billiards with Corner Points

It is known that the dynamics of planar billiards satisfies strong mixing properties (e.g., exponential decay of correlations), provided that some expansion condition on unstable curves is satisfied. This condition has been shown to always hold for smooth dispersing planar billiards, but it needed to be assumed separately in the case of dispersing planar billiards with corner points. We prove that this expansion condition holds for any dispersing planar billiard with corner points, no cusps and bounded horizon.     

A quantified Tauberian theorem for the Laplace-Stieltjes transform

We study the general rotation sets of open billiards in \({\mathbb {R}}^2\) for the observable given by the starting point of a given billiard trajectory. We prove that, for a class of open billiards, the general rotation set is equal to the polygon formed by the midpoints of the segments connecting the centers of the obstacles. Moreover, we provide an example to show that such a result may not hold in general.     

Modeling Nondegenerate Bifurcations of Closures of Solutions for Integrable Systems with Two Degrees of Freedom by Integrable Topological Billiards

It is well known that surgeries of closures of solutions for integrable nondegenerate Hamiltonian systems with two degrees of freedom at a level of constant energy are classified by the so-called 3-atoms. These surgeries correspond to singular leaves of the Liouville foliation of three-dimensional isoenergetic surfaces. In this paper we prove the Fomenko conjecture that all such surgeries are modeled by integrable topological two-dimensional billiards (billiard books).     

Creating Transverse Homoclinic Connections in Planar Billiards

Given a planar billiard system containing stable and unstable manifolds that intersect nontransversely, we show how to make a local perturbation to the boundary that causes the intersection to become transverse. We apply these ideas to billiards inside an ellipse. Bibliography: 19 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 122–134.his revised version was published in June 2005 with corrections in the author affiliation.     

Billiards and Nonholonomic Distributions

In this note, we consider billiards with full families of periodic orbits. It is shown that the construction of a convex billiard with a “rational” caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to an (N - 1)-dimensional distribution on a (2N - 1)-dimensional manifold. We describe the properties of this distribution, as well as some important consequences for billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist. Bibliography: 13 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 56–64.     

Two-link billiard trajectories: extremal properties and stability

Two-link periodic trajectories of a plane convex billiard, when a point mass moves along a segment which is orthogonal to the boundary of the billiard at its end points, are considered. It is established that, if the caustic of the boundary lies within the billiard, then, in a typical situation, there is an even number of two-link trajectories and half of them are hyperbolic (and, consequently, unstable) and the other half are of elliptic type. An example is given of a billiard for which the caustic intersects the boundary and all of the two-link trajectories are hyperbolic. The analysis of the stability is based on an analysis of the extremum of a function of the length of a segment of a convex billiard which is orthogonal to the boundary at one of its ends.     

Weyl-flow and the conformally symplectic structure of thermostatted billiards: The problem of the hyperbolicity

The highly complex nature of the transport in thermostatted billiards has been of interest in the last few decades because of industrial and medical applications. The onset of hyperbolic dynamics (deterministic chaos) in such a billiard has evidenced an interesting stabilization of the transport properties, especially in microporous media. Recently, different mathematical methods have been developed for establishing hyperbolicity in thermostatted billiards, among these, the Weyl-flow and the conformally symplectic structure techniques.This paper deals with analytical investigations on the possible hyperbolic nature of two thermostatted billiards: The nonequilibrium Ehrenfest gas (NEEG) and the pump model (PM). Despite numerical investigations supporting the idea of their dissipative dynamics, the hyperbolicity of these billiards has not been yet established. The analysis developed in this paper shows how the Weyl-flow technique has failed for NEEG, revealing the necessity to develop new strategies in order to obtain hyperbolicity. On the contrary, we prove that the PM has a conformally symplectic structure, which is the basis for establishing the hyperbolicity of such a hybrid dynamical system.