Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures are living in the (commutative) vector space $\real^n$, if the toroidal billiard is given on the flat $n$-torus.
The billard trajectories, being curves, oftentimes getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e., the billiard flow on the 2-torus with a single, convex obstacle removed. This fundamental group is known to be the group $\textbf{F}_2$ freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov's sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classicaly expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits.
The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(\textbf{F}_2)$ of all ``ends'' of the hyperbolic group $\textbf{F}_2$. An element of $\text{Ends}(\textbf{F}_2)$ describes the direction in (the Cayley graph of) the group $\textbf{F}_2$ in which the considered trajectory escapes to infinity, whereas the height function $t$ ($t \ge 0$) of the cone gives us the average speed at which this escape takes place.
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 $P$ which contain a periodic billiard orbit agrees with the analogous set for $Q$. We study this equivalence relationship and compare it to the equivalence relations, order equivalence and code equivalence, introduced in \cite{BT1,BT2}. 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…
Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. The billard trajectories, being curves, oftentimes getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e., the billiard flow on the 2-torus with a single, convex obstacle removed. This fundamental group is known to be the group $\textbf{F}_2$ freely generated by two elements, w…
In this paper we study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with three, mutually intersecting and mutually orthogonal cylindrical scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(\textbf{F}_3)$ of all "ends" of the hyperbolic group $\textbf{F}_3=π_1(\mathbf{Q})$. An element of $\text{Ends}(\textbf{F}_3)$ describes the direction in (the Cayley graph of) the group $\textbf{F}_3$ in which the considered trajectory escapes to infinity, whereas th…
We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two, disjoint and orthogonal cylindrical scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(G)$ of all "ends" of the hyperbolic group $G=π_1(\mathbf{Q})$. An element of $\text{Ends}(G)$ describes the direction in (the Cayley graph of) the group $G$ in which the considered trajectory escapes to infinity, whereas the height function $s$ ($s\ge 0$) of the cone gives us the average speed at which t…
We introduce the concepts of rotation numbers and rotation vectors for billiard maps. Our approach is based on the birkhoff ergodic theorem. We anticipate that it will be useful, in particular, for the purpose of establishing the non-ergodicity of billiard in certain domains.
We investigate the rotation sets of open billiards in $\mathbb{R}^N$ for the natural observable related to a starting point of a given billiard trajectory. We prove that the general rotation set is convex and the set of all convex combinations of rotation vectors of periodic trajectory $P_ϕ$ is dense in it. We provide a constructive proof which illustrates that the set $P_ϕ$ is dense in the pointwise rotation set, and the closure of the pointwise rotation set is convex. We also consider a class of billiards consisting of three obstacles and construct a sequence in the symbol space such that it…
In this paper, we discuss rotation number on the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon $η$, we can construct an outer billiard table $T$ by cutting out a fixed area from the interior of $η$. $T$ is piece-wise hyperbolic and the polygon $η$ is an invariant curve of $T$ under the billiard map $ϕ$. We will show that, if $β$ is a periodic point under the outer billiard map with rational rotation number $τ= p / q$, then the $n$th iteration of the billiard map is not the local identity at $β$. This proves that the rotation number $τ$ as a function…
For billiards with $N$ obstacles on a torus, we study the behavior of specific kind of its trajectories, \emph{the so called admissible trajectories}. Using the methods developed in \cite{1}, we prove that the \emph{admissible rotation set} is convex, and the periodic trajectories of admissible type are dense in the admissible rotation set. In addition, we show that the admissible rotation set is a proper subset of the general rotation set.
We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ homotopic to the identity. A set of hypotheses is presented, called fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincaré disk $\mathbb{D}),$ denoted $\widetilde{f},$ has complicated and rich dynamics. In this context we generalize results that hold for homeomorphisms of the torus homotopic to the identity when their rotation sets contain zero in the interior. In particular, we prove that if $f$…
This article is concerned with the study of Mather's β-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum. After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both th…
We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets.
Rigidity of Mather's $β$-function on a KAM set for analytic billiards-like maps and unique quasi-analytic continuation
Università degli Studi di Roma Tor Vergata（罗马托尔维加塔大学）；Institute of Science and Technology Austria（奥地利科学技术研究所）；Centre de Mathématiques Laurent Schwartz (CMLS), CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院高等研究院CNRS洛朗·施瓦茨数学中心）
For a given rotation number we compute the Hausdorff dimension of the set of well approximable numbers. We use this result and an inhomogeneous version of Jarnik's theorem to show strong recurrence properties of the billiard flow in certain polygons
We show that for a fixed curve $K$ and for a family of variables curves $L$, the number of $n$-Poncelet pairs is $\frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd $ (m,n)=1$. The curvee $K$ do not have to be part of the family. In order to show this result we consider an associated billiard transformation and a twist map which preserves area. We use Aubry-Mather theory and the rotation number of invariant curves to obtain our main result. In the last section we estimate the derivative of the rotation number of a general twist map usin…
Translation and rotation numbers have played an interesting and important role in the qualitative description of various dynamical systems. In this exposition we are especially interested in applications which lead to proofs of periodic motions in various kinds of dynamics on the annulus. The applications include billiards and geodesic flows. Going beyond this simple qualitative invariant in the study of the dynamics of area preserving annulus maps, G.D. Birkhoff was led to the concept of ``regions of instability'' for twist maps. We discuss the closely related notion of instability sets for a…
We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in a limit when the number of iter…
A comprehensive study of periodic trajectories of billiards within ellipsoids in $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line. By leveraging deep, but yet not widely known results of the Krein-Levin-Nudelman theory of generalized Chebyshev polynomials, fundamental properties of billiard dynamics are proven for any $d$, viz., that the sequences of winding numbers are monotonic. By employing the potential theory we pr…
We consider a one-parameter family of invertible maps of a two-dimensional lattice, obtained by discretising the space of planar rotations. We let the angle of rotation approach $π/2$, and show that the limit of vanishing discretisation is described by an integrable piecewise-smooth Hamiltonian flow, whereby the plane foliates into families of invariant polygons with an increasing number of sides. Considered as perturbations of the flow, the lattice maps assume a different character, described in terms of strip maps, a variant of those found in outer billiards of polygons. The perturbation int…
A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line and ellipsoidal billiards in $d$-dimensional space. Even in the planar case, systematically studied in the present paper it leads to new results in characterizing $n$ periodic trajectories vs. so-called $n$ elliptic periodic trajectories, which are $n$-periodic in elliptical coor…
