A couple of days ago (on November 12th, 2014 to be more precise), Giovanni Forni gave a talk at the “flat seminar / séminaire plat” on the ergodicity of billiards on non-rational polygons, and, by following the suggestion of two friends, I will transcript in this post my notes from Giovanni’s talk.
[Update (November 20, 2014): Some phrases near the statement of Theorem 3 below were edited to correct an inaccuracy pointed out to me by Giovanni.]
Let be a polygon with sides and denote by its interior angles.
The billiard flow associated to is the following dynamical system. A point-particle in follows a linear trajectory with unit speed until it hits the boundary of . At such an instant, the point-particle is reflected by the boundary of (according to the usual laws of a specular reflection) and then it follows a new linear trajectory with unit speed. (Of course, this definition makes no sense at the corners of , and, for this reason, we leave the billiard flow undefined at any orbit going straight into a corner)
The phase space of the billiard flow is naturally identified with the three-dimensional manifold : indeed, we need an element of to describe the position of the particle and an element of the unit circle to describe the velocity vector of the particle.
Alternatively, the billiard flow associated to can be interpreted as the geodesic flow on a sphere with a flat metric and conical singularities (whose cone angles are ) with non-trivial holonomy (see Section 2 of Zorich’s survey): roughly speaking, one obtains this flat sphere with conical singularities by taking two copies of (one on the top of the other), gluing them along the boundaries, and by thinking of a billiard flow trajectory on as a straight line path going from one copy of to the other at each reflection.
This interpretation shows us that billiard flows on polygons are a particular case of geodesic flows on the unit tangent bundle of compact flat surfaces whose subsets of conical singularities were removed.
Remark 1 In the case of a rational polygon (i.e., are rational multiples of ), it is often a better idea (see this survey of Masur and Tabachnikov) to take several copies of obtained by applying the finite group generated by the reflections through the sides of and then glue by translation the pairs of parallel sides of the resulting figure. In this way, one obtains that the billiard flow associated to is equivalent to translation (straightline) flow on a translation surface (an object that has trivial holonomy and, hence, is more well-behaved that a flat metric on with conical singularities) and this partly explains why the Ergodic Theory of billiards on rational polygons is well-developed. However, let us not insist on this point here because in what follows we will be mostly interested in billiard flows on irrational polygons.
A basic problem concerning the dynamics of billiards flows on polygons, or, more generally, geodesic flows on flat surfaces with conical singularities is to determine whether such a dynamical system is ergodic.
In view of Remark 1, we can safely skip the case of rational polygons: indeed, this setting one can use the relationship to translation surfaces to give a satisfactory answer to this problem (see the survey of Masur and Tabachnikov for more explanations). So, from now on, we will focus on billiard flows associated to non-rational polygons.
Kerckhoff, Masur and Smillie proved in 1986 that the billiard flow is ergodic for a -dense subset of polygons. Their idea is to consider the -dense subset of “Liouville polygons” admitting fast approximations by rational polygons (i.e., the subset of polygons whose interior angles admit fast approximations by rational multiples of ). Because the ergodicity of the billiard flow on rational polygons is well-understood, one can hope to “transfer” this information from rational polygons to any “Liouville polygon”.
Remark 2 The -dense subset of polygons constructed by Kerckhoff, Masur and Smillie has zero measure: indeed, this happens because they require the angles to be “Liouville” (i.e., admit fast approximations by rational multiples of ), and, as it is well-known, the subset of Liouville numbers has zero Lebesgue measure.
A curious feature of the argument of Kerckhoff, Masur and Smillie is that it is hard to extract any sort of quantitative criterion. More precisely, it is difficult to quantify how fast the quantities must be approximated by rationals in order to ensure that the ergodicity of the billiard flow on the corresponding polygon. This happens because the genera of translation surfaces associated to the rational polygons approximating usually tend to infinity and it is a non-trivial problem to control the ergodic properties of translation flows on families of translation surfaces whose genera tend to infinity.
Nevertheless, Vorobets obtained in 1997 (by other methods) a quantitative version of Kerckhoff, Masur and Smillie by showing the ergodicity of the billiard flow on a polygon whose interior angles verify the following fast approximation property: there exist arbitrarily large natural numbers such that
for some rational numbers , , with denominators , .
In summary, the works of Kerckhoff-Masur-Smillie and Vorobets allows to solve the problem of ergodicity of the billiard flow on Liouville polygons.
Of course, this scenario motivates the question of ergodicity of billiard flows on Diophantine polygons (i.e., the “complement” of Liouville polygons consisting of those which are badly approximated by rational polygons).
In his talk, Giovanni announced a new criterion for the ergodicity of the billiard flow on polygons (and, more generally, the geodesic flow on a flat surface with conical singularities) with potential applications to a whole class (of full measure) of Diophantine polygons.