# Existence of nonergodic polygonal billiard

Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one.

A standard conjecture is that a typical billiard flow is ergodic. But are there examples of irrational polygonal billiards for which the flow is known not to be ergodic ?

EDIT : It is obvious that in this sense every rational polygon is not ergodic. The good framework is irrational polygons, for which the question seems to be open.

E. Gutkin, Billiard dynamics: An updated survey with the emphasis on open problems Chaos 22, 026116 (2012) gives as open problem 8: Give an example of an irrational but non-ergodic polygon. Some recent numerical results are given in J. Wang et al, Non-ergodicity and localization of invariant measure for two colliding masses arxiv:1309.7617.

For rational polygons, there is a nice example of an L-shaped billiard and a non-ergodic direction in Fig 3 of L. Demarco The conformal geometry of billiards Bull. AMS 48 33-52 (2011).

Update: There is also literature on the somewhat related question of non-dense aperiodic orbits. If a positive measure of orbits is not dense, the billiard is not ergodic. G. A. Galpern Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons Commun. Math. Phys. 91 187-211 (1983) gives examples of polygons with $n\geq 3$, including with irrational angles, which have non-dense aperiodic orbits. G. W. Tokarsky Galpern's triangle example Commun. Math. Phys. 335 1211–1213 (2015) shows that the $n=3$ example orbit is in fact periodic, so the existence of a non-dense aperiodic orbit in a triangular billiard appears to be still open.

• You are here dealing with non ergodic direction. Do you know if anything is known about the action on the phase space ? – Selim G Apr 17 '14 at 14:30
• @SelimG I'm not sure exactly what you are asking. In a rational billiard, each trajectory visits only a finite number of directions, so is far from uniform in momentum space. By "non-ergodic direction" I mean a direction in which the orbit is non-uniform in its available phase space: position space and the finite set of directions. – user25199 Apr 17 '14 at 14:46
• Yes, that is why I changed my question to irrational billiards :) Actually my question is exactly Gutkin's. Thanks for the link by the way. – Selim G Apr 17 '14 at 14:55

The dynamics of billiard flows in rational polygons (J. Smillie, 2000): The billiard in a rational polygon is ergodic in "almost all" directions, more precisely, the Hausdorff dimension of the set of non-ergodic directions is at most 1/2.

• An explicit example of Hausdorff dimension 1/2 may be found in Y. Cheung, "Hausdorff Dimension of the Set of Nonergodic Directions", Ann Math 158, 661-678 (2003). – user25199 Apr 17 '14 at 12:47

There are examples in the slit torus. Prof. Smillie explains this in: http://www2.warwick.ac.uk/fac/sci/maths/people/staff/john_smillie/course/notes14february.pdf

• As for Carl, Smillie seems to deal with specific directions. I was wondering about the action on the phase space. – Selim G Apr 17 '14 at 14:31

Numerical evidences for non-ergodic irrational triangles may be found in

Lima, Rodríguez-Pérez and de Aguiar, Phys. Rev. E 87, 062902 (2013).