Numerical exploration of a family of strictly convex billiards with boundary of classC 2

We are interested in the possible existence of strictly convex ergodic billiards. Such billiards are searched for by means of numerical investigation. The boundary of a billiard is built with four arcs of classC . Adjacent arcs have equal curvatures at connecting points. The surface of section of the billiards is explored. It seems as if symmetric billiards always have invariant curves (islands). Asymmetric billiards have been found which look ergodic. They are built with an arc of an ellipse, two arcs of circles, and one-half of a Descartes oval.     

Using integrability to produce chaos: Billiards with positive entropy

A new open class of convex 2 dimensional planar billiards with positive Lyapunov exponent almost everywhere is constructed. We introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive Lyapunov exponents. We prove that under smallC ⁶ perturbations, focusing arcs remain focusing and thereby show that perturbations of the Bunimovich stadium billiard have positive Lyapunov exponents.Partially supported by NSF grant DMS 8806067     

Synchronization Properties of Random Piecewise Isometries

Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table. In the present paper we demonstrate that a class of convex tables—asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. This result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than 40 years ago.     

Numerical experiments on billiards

We investigate decay properties of correlation functions in a class of chaotic billiards. First we consider the statistics of Poincaré recurrences (induced by a partition of the billiard): the results are in agreement with theoretical bounds by Bunimovich, Sinai, and Bleher, and are consistent with a purely exponential decay of correlations out of marginality. We then turn to the analysis of the velocity-velocity correlation function: except for intermittent situations, the decay is purely exponential, and the decay rates scale in a simple way with the (uniform) curvature of the dispersing arcs. A power-law decay is instead observed when the system is equivalent to an infinite-horizon Lorentz gas. Comments are given on the behaviour of other types of correlation functions, whose decay, during the observed time scale, appears slower than exponential.     

Semi-Focusing Billiards: Hyperbolicity

In this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of ℝ³. We also prove that a related class of semi-focusing billiards has mixed dynamics, i.e., their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other. These billiards are the first rigorous examples of billiards in domains of ℝ³ with divided phase space. The first author was partially supported by the NSF grant #0140165 and the Humboldt Foundation. The second author was partially supported by the FCT (Portugal) through the Program POCTI/FEDER.     

Local instability of orbits in polygonal and polyhedral billiards

We classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons, for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.     

One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.     

Principles for the design of billiards with nonvanishing Lyapunov exponents

We introduce a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents.     

Billiards: a singular perturbation limit of smooth Hamiltonian flows

Nonlinear multi-dimensional Hamiltonian systems that are not near integrable typically have mixed phase space and a plethora of instabilities. Hence, it is difficult to analyze them, to visualize them, or even to interpret their numerical simulations. We survey an emerging methodology for analyzing a class of such systems: Hamiltonians with steep potentials that limit to billiards.     

The influence of confining wall profile on quantum interference effects in etched Ga0.25In0.75As/InP billiards

We present measurements of the potential profile of etched GaInAs/InP billiards and show that their energy gradients are an order of magnitude steeper than those of surface-gated GaAs/AlGaAs billiards. Previously observed in GaAs/AlGaAs billiards, fractal conductance fluctuations are predicted to be critically sensitive to the billiard profile. Here we show that, despite the increase in energy gradient, the fractal conductance fluctuations persist in the harder GaInAs/InP billiards.     

Negative Magneto-Resistance Beyond Weak Localization in Three-Dimensional Billiards:Effect of Arnold Diffusion

We investigate a semiclassical conductance for ballistic open three-dimensional (3-d) billiards. For partially or completely broken-ergodic 3-d billiards such as SO(2) symmetric billiards, the dependence of the conductance on the Fermi wavenumber is dramatically changed by the lead orientation. Application of a symmetry-breaking weak magnetic field brings about mixed phase-space structures of 3-d billiards which ensures a novel Arnold diffusion that cannot be seen in 2-d billiards. In contrast to the 2-d case, the anomalous increment of the conductance should inevitably include a contribution arising from Arnold diffusion as well as a weak localization correction. Discussions are devoted to the physical condition for observing this phenomenon.     

Nonequilibrium Gas and Generalized Billiards

Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.     

The K-property of “orthogonal” cylindric billiards

Toric billiards with cylindric scatterers (briefly cylindric billards) generalize the class of Hamiltonian systems of elastic hard balls. In this paper a class of cylindric billiards is considered where the cylinders are orthogonal or more exactly: the constituent space of any cylindric scatterer is spanned by some of the (of course, orthogonal) coordinate vectors adapted to the euclidean torus. It is shown that the natural necessary condition for the K-property of such billiards is also sufficient.Research supported by the Hungarian National Foundation for Scientific Research, grant No. 1902     

Rashba billiards

We study the energy levels of non-interacting electrons confined to move in two-dimensional billiard regions and having a spin-dependent dynamics due to a finite Rashba spin splitting. The free space Green's function for such Rashba billiards is constructed analytically and used to find the area and perimeter contributions to the density of states, as well as the corresponding smooth counting function. We show that, in contrast to systems with spin-rotational invariance, Rashba billiards always possess a negative energy spectrum. A semi-classical analysis is presented to interpret the singular behavior of the density of states at certain negative energies for circular Rashba billiards. Our detailed analysis of the spin structure of circular Rashba billiards reveals a finite out-of-plane spin projection for electron eigenstates.     

Chaos in composite billiards

The mechanisms and features of the chaotic behavior in billiards with ray splitting (refraction) are considered. In contrast to ordinary billiards, the law of motion in composite billiards that is coded with a sequence of ray visits to different media is shown to be deterministically chaotic. The analysis is performed in terms of a geometrical-dynamical approach in which a symmetric phase space is used instead of the ordinary Hamiltonian phase space. The chaotization elements in composite billiards of a general position are studied. The dynamics of rays in ring billiards consisting of two concentric media with different refractive indices is considered.     

Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

Nonergodicity of the motion in three-dimensional steep repelling dispersing potentials

It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems that are arbitrarily close to three-dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are nonergodic. The mechanism for creating the islands is corners of the billiards domain.     

Scars of periodic orbits in the stadium action billiard

Compact billiards in phase space, or action billiards, are constructed by truncating the classical Hamiltonian in the action variables. The corresponding quantum mechanical system has a finite Hamiltonian matrix. In previous papers we defined the compact analog of common billiards, i.e., straight motion in phase space followed by specular reflections at the boundaries. Computation of their quantum energy spectra establishes that their properties are exactly those of common billiards: the short-range statistics follow the known universality classes depending on the regular or chaotic nature of the motion, while the long-range fluctuations are determined by the periodic orbits. In this work we show that the eigenfunctions also follow qualitatively the general characteristics of common billiards. In particular, we show that the low-lying levels can be classified according to their nodal lines as usual and that the high excited states present scars of several short periodic orbits. Moreover, since all the eigenstates of action billiards can be computed with great accuracy, Bogomolny's semiclassical formula for the scars can also be tested successfully.