# Poincaré last theorem

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

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

and 3) the points with move counter-clockwise and the points with clockwise, that is, , . 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 géometrie" 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)