Billiard Flow Research Articles

Algebraic non-integrability of magnetic billiards

We consider billiard ball motion in a convex domain of the Euclidean plane bounded by a piece-wise smooth curve under the action of a constant magnetic field. We show that if there exists a first integral polynomial in the velocities of the magnetic billiard flow, then every smooth piece γ of the boundary must be algebraic, and either is a circle or satisfies very strong restrictions. In particular, it follows that any non-circular magnetic Birkhoff billiard is not algebraically integrable for all but finitely many values of the magnitude of the magnetic field. Moreover, a magnetic billiard in ellipse is not algebraically integrable for all values of the magnitude of the magnetic field. We conjecture that the circle is the only integrable magnetic billiard, not only in the algebraic sense, but also for a broader meaning of integrability. We also introduce what we call outer magnetic billiards. As an application of our method, we prove analogous results on algebraically integrable outer magnetic billiards.

Wigner measures and observability for the Schrödinger equation on the disk

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a “structure theorem”, expressing the restriction of the Wigner measures on each invariant torus in terms of second-microlocal measures. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty principle, and are shown to propagate according to Heisenberg equations on the circle. Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the $$L^2$$ -norm of a solution on any open subset intersecting the boundary (resp. the $$L^2$$ -norm of the Neumann trace on any nonempty open set of the boundary) controls its full $$L^2$$ -norm (resp. $$H^1$$ -norm). These results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary.

Anosov Geodesic Flows, Billiards and Linkages

Any smooth surface in R^3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R^2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive and has finite horizon, then the geodesic flow of the corresponding surface is Anosov. We apply this result to the theory of mechanical linkages and their dynamics: we provide a new example of a simple linkage whose physical behavior is Anosov. For the first time, the edge lengths of the mechanism are given explicitly.

Obstacles with non-trivial trapping sets in higher dimensions

Using a well-known example of Livshits, for every \({n > 2}\) we construct obstacles K in \({\mathbb{R}^n}\) such that the set of trapped points for the billiard flow in the exterior of K has a non-empty interior, and therefore a positive Lebesgue measure.

A Franks' lemma for convex planar billiards

Let γ be an orbit of the billiard flow on a convex planar billiard table; then the perpendicular part of the derivative of the billiard flow along γ is a symplectic linear map DP. This paper contains a proof of the following Franks’ lemma for a residual set of convex planar billiard tables: for any closed orbit, the map DP can be perturbed freely within a neighbourhood in Sp(1) by a C2-small perturbation in the space of convex planar billiard tables.

A Thermodynamic Formalism Approach to the Selberg Zeta Function for Hecke Triangle Surfaces of Infinite Area

We provide an explicit construction of a cross section for the geodesic flow on infinite-area Hecke triangle surfaces which allows us to conduct a transfer operator approach to the Selberg zeta function. Further we construct closely related cross sections for the billiard flow on the associated triangle surfaces and endow the arising discrete dynamical systems and transfer operator families with two weight functions which presumably encode Dirichlet respectively Neumann boundary conditions. The Fredholm determinants of these transfer operator families constitute dynamical zeta functions, which provide a factorization of the Selberg zeta function of the Hecke triangle surfaces.

Topological Mixing for Some Residual Sets of Interval Exchange Transformations

We show that a residual set of non-degenerate IETs on more than 3 letters is topologically mixing. This shows that there exists a uniquely ergodic topologically mixing IET. This is then applied to show that some billiard flows in a fixed direction in an L-shaped polygon are topologically mixing.

Geodesic and Billiard Flows on Quadrics in Pseudo-Euclidean Spaces: L–A Pairs and Chasles Theorem

In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the integrability analogous to the classical Chasles theorem for symmetric ellipsoids is given. We also consider a generalization of the billiard within arbitrary quadric allowing virtual billiard reflections.

Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow

By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families:$$\begin{eqnarray}Z(s)=\det (1-{\mathcal{L}}_{s}^{+})\det (1-{\mathcal{L}}_{s}^{-}).\end{eqnarray}$$In this article we show that the operator families${\mathcal{L}}_{s}^{\pm }$arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for$s\in \mathbb{C}$,$\text{Re}s={\textstyle \frac{1}{2}}$, the operator${\mathcal{L}}_{s}^{+}$(respectively${\mathcal{L}}_{s}^{-}$) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue$s(1-s)$. For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.

Homotopical rigidity of polygonal billiards

Consider two k-gons P and Q. We say that the billiard flows in P and Q are homotopically equivalent if the set of conjugacy classes in the fundamental group of P, viewed as a punctured sphere, which contain a periodic billiard orbit agrees with the analogous set for Q. We study this equivalence relationship and compare it to the notions of order equivalence and code equivalence, introduced in [1,2]. In particular we show if P is a rational polygon, and Q is homotopically equivalent to P, then P and Q are similar, or affinely similar if all sides of P are vertical and horizontal.

Regular simplices and periodic billiard orbits

A simplex is the convex hull of $n+1$ points in $\mathbb{R}^{n}$ which form an affine basis. A regular simplex $\Delta^n$ is a simplex with sides of the same length. We consider the billiard flow inside a regular simplex of $\mathbb{R}^n$. We show the existence of two types of periodic trajectories. One has period $n+1$ and hits once each face. The other one has period $2n$ and hits $n$ times one of the faces while hitting once any other face. In both cases we determine the exact coordinates for the points where the trajectory hits the boundary of the simplex.

Ergodic Directions for Billiards in a Strip with Periodically Located Obstacles

We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<\lambda<h$. We prove that the set of ergodic directions is always uncountable. Moreover, if $\lambda/h\in(0,1)$ is rational the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.

Quantum ergodicity for a class of mixed systems

We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrödinger operators with mixed assumptions on the Hamiltonian flow.) Specifically, we assume that the billiard flow has an invariant ergodic component, U , and study defect measures, \mu , of positive density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence satisfies \mu|_{U}=c\mu_L|_{U} , where \mu_L is the Liouville measure. This proves part of a conjecture of Percival [18].

Pseudo-integrable billiards and arithmetic dynamics

We introduce a new class of billiard systems in the plane, withboundaries formed by finitely many arcs of confocal conics such thatthey contain some reflex angles. Fundamental dynamical, topological,geometric, and arithmetic properties of such billiards are studied.The novelty, caused by reflex angles on boundary, induces invariantleaves of higher genera and dynamical behavior different fromLiouville--Arnold's Theorem. Its analog is derived from the MaierTheorem on measured foliations. The billiard flow generates ameasurable foliation defined by a closed 1-form $w$. Using theclosed form, a transformation of the given billiard table to arectangular cylinder is constructed and a trajectory equivalencebetween corresponding billiards has been established. A local versionof Poncelet Theorem is formulated and necessary algebro-geometricconditions for periodicity are presented. It is proved that thedynamics depends on arithmetic of rotation numbers, but not ongeometry of a given confocal pencil of conics.

Non-ergodic $\mathbb{Z}$ -periodic billiards and infinite translation surfaces

We give a criterion which proves non-ergodicity for certain infinite periodic billiards and directional flows on $\mathbb{Z}$ -periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a $\mathbb{Z}^{2}$ -periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding directional billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent $\mathbb{Z}$ -cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skew-products which arise as Poincaré maps of the above systems are associated to non-regular $\mathbb{Z}$ -valued cocycles for interval exchange transformations.

Singularities and non-hyperbolic manifolds do not coincide

We consider the billiard flow of elastically colliding hard balls on the flat ν-torus (ν ⩾ 2), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli mixing property) of all such systems, i.e. the verification of the Boltzmann–Sinai ergodic hypothesis.

Divergent trajectories in the periodic wind-tree model

The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S. Leli\`evre and S. Troubetzkoy (arXiv:0912.2891v1) that for a residual set of parameters (a,b) the billiard flow in T(a,b) is recurrent in almost every direction. We prove that for many parameters (a,b) there exists a set S of angles of positive Hausdorff dimension such that every billiard trajectory in T(a,b) with initial angle in S is self-avoiding. In particular, the flow in a direction of S is divergent.

Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards

The Koch snowflake KS is a nowhere differentiable curve. The billiard table Omega(KS) with boundary KS is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes Omega(KS) such an interesting, yet difficult, table to analyze. In this paper, we approach this problem by approximating (from the inside) Omega(KS) by well-defined (prefractal) rational polygonal billiard tables Omega(KS_n). We first show that the flat surface S(KS_n) determined from the rational billiard Omega(KS_n) is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [Gut2], allows us to define a sequence of compatible orbits of prefractal billiards. We define a hybrid orbit of a prefractal billiard Omega(KS_n) and show that every dense orbit of a prefractal billiard is a dense hybrid orbit of Omega(KS_n). This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits. We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of Omega(KS). In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. Finally, we discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining `fractal flat surfaces' naturally associated with the billiard flows.

Parking garages with optimal dynamics

We construct generalized polygons (‘parking garages’) in which the billiard flow satisfies the Veech dichotomy, although the associated translation surface obtained from the Zemlyakov–Katok unfolding is not a lattice surface. We also explain the difficulties in constructing a genuine polygon with these properties.

Distribution of Periods of Closed Trajectories in Exponentially Shrinking Intervals

For hyperbolic flows over basic sets we study the asymptotic of the number of closed trajectories γ with periods T γ lying in exponentially shrinking intervals $${(x - e^{-\delta x}, x + e^{-\delta x}), \; \delta > 0, \; x \to + \infty.}$$ A general result is established which concerns hyperbolic flows admitting symbolic models whose corresponding Ruelle transfer operators satisfy some spectral estimates. This result applies to a variety of hyperbolic flows on basic sets, in particular to geodesic flows on manifolds of constant negative curvature and to open billiard flows.