View source for Poincaré last theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730601.png" /> be an annulus in the plane bounded by circles with radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730602.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730603.png" /> and let a mapping from this domain onto itself (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730604.png" /> is the polar angle), given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730605.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730606.png" /></td> </tr></table>

and 3) the points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730607.png" /> move counter-clockwise and the points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730608.png" /> clockwise, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p0730609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073060/p07306010.png" />. 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 géometrie"  ''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>