View source for Poincaré last theorem

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

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

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