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There is a conjecture by Birkhoff which claims that for a simple closed $C^2$ plane curve $C$, if the billiard ball map is integrable then the curve is an ellipse.

Integrability here might be formulated as follows: there exists a neighbourhood of $C$ in the interior $Int(C)$ that is foliated by caustics (caustics being curves that are everywhere tangent to a given trajectory of the billiard ball).

I would be interested to know the current status (and progresses, if there are) of this conjecture.

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    $\begingroup$ Currently there is no Wikipedia article on the Birkhoff conjecture. Would someone here care to remedy this deficiency? $\endgroup$
    – Michael Ruxton
    Commented Nov 14, 2019 at 0:35

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I haven't heard of any recent breakthroughs. The strongest result that I know is due to Misha Bialy:

Theorem. If almost every phase point of the billiard ball map in a strictly convex billiard table belongs to an invariant circle, then the billiard table is a disc.

Stronger results are available for an outer version of the Birkhoff conjecture. Tabachnikov proved that if the outer billiard map around a plane oval is algebraically integrable then the oval is an ellipse (article, arXiv version).

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    $\begingroup$ Then what one should think about the following paper? nyjm.albany.edu/j/1997/3-2.html The author claims that he has solved the conjecture for $C^\infty$ curves. $\endgroup$
    – DamienC
    Commented Aug 13, 2010 at 19:37
  • $\begingroup$ Unfortunately most people I talked to don't believe Amiran's proof is correct. I don't know what exactly is wrong with it. $\endgroup$
    – Eugene Lerman
    Commented Aug 30, 2012 at 15:03
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For a recent progress see http://arxiv.org/pdf/1412.2853.pdf a local version of this conjecture is proven.

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I'm no expert, but according to Tabachnikov the conjecture was still open as of 2005, while Delshams and Ramirez-Ros have a local result (i.e. the conjecture is true when considering symmetric entire perturbations). Probably Mathscinet would help more.

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    $\begingroup$ (By the way, hello Damien!) $\endgroup$
    – Thomas Sauvaget
    Commented Jul 5, 2010 at 20:19
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I think has recently obtained partial results on this conjecture (On polynomially integrable Birkhoff billiards on surfaces of constant curvature, JEMS 2021)

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