Poincaré last theorem

Let $ K $ be an annulus in the plane bounded by circles with radii $ r = a $ and $ r = b $ and let a mapping from this domain onto itself ( $ \theta $ is the polar angle), given by

$$ \widetilde{r} = \phi ( r , \theta ) ,\ \ \widetilde \theta = \psi ( r , \theta ) , $$

satisfy the conditions: 1) the mapping preserves area; 2) each boundary circle maps onto itself,

$$ \phi ( a , \theta ) = a ,\ \ \phi ( b , \theta ) = b ; $$

and 3) the points with $ r = a $ move counter-clockwise and the points with $ r = b $ clockwise, that is, $ \psi ( a , \theta ) > \theta $, $ \psi ( b , \theta ) < \theta $. Then this mapping has two fixed points. More generally, instead of preserving area one can require that no subdomain maps to a proper subset of itself.

This theorem was stated by H. Poincaré [1] in 1912 in connection with certain problems of celestial mechanics; it was proved by him in a series of particular cases but he did not, however, obtain a general proof of this theorem. The paper was sent by Poincaré to an Italian journal (see [1]) two weeks before his death, and the author expressed his conviction, in an accompanying letter to the editor, of the validity of the theorem in the general case.

References

[1]	H. Poincaré, "Sur un théorème de géométrie" Rend. Circ. Mat. Palermo , 33 (1912) pp. 375–407
[2]	G. Birkhoff, "Proof of Poincaré's geometric theorem" Trans. Amer. Math. Soc. , 14 (1913) pp. 14–22
[3]	L.A. Pars, "A treatise on analytical dynamics" , Heinemann , London (1965)

Comments

A proof of Poincaré's last theorem is in [2]. It is also known as the Poincaré–Birkhoff fixed-point theorem.

References

[a1]	V.I. Arnol'd, A. Avez, "Problèmes ergodiques de la mécanique classique" , Gauthier-Villars (1967) pp. §20.5; Append. 29 (Translated from Russian)
[a2]	G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927)