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.


[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)


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


[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)
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Poincaré last theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article