Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
A related question and answer on math.SE: math.stackexchange.com/questions/6038/… –  Carl Apr 17 '14 at 13:26

3 Answers 3

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.

share|improve this answer
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). –  Carl Apr 17 '14 at 12:47

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).

share|improve this answer
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. –  Carl 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

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

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.