Poincaré-Birkhoff's theorem

The set of Poincaré-Birkhoff (also Poincaré's last sentence or Poincaré Geometric set is called) is a theorem of the symplectic geometry and the theory of dynamical systems on the existence of fixed points for area-preserving mappings. It states that an area-preserving image of a circular ring , which rotates the two edge components in different directions, must have at least two fixed points.

Sometimes a conclusion is also called the Poincaré-Birkhoff theorem. It says that when a dynamic system with invariant tori of rational number of turns is perturbed, an even number of fixed points of the Poincaré map is retained.

One of the higher-dimensional generalizations of the Poincaré-Birkhoff theorem is the Arnold conjecture, proven by means of Floer's homology .

sentence

Be with and be the circular ring ${\ displaystyle a, b \ in \ mathbb {R} ^ {+}}$ ${\ displaystyle a <b}$ ${\ displaystyle R}$

{\ displaystyle R = \ left \ {(x, y) \ colon (x, y) \ in \ mathbb {R} ^ {2}, a \ leq {\ sqrt {x ^ {2} + y ^ {2 }}} \ leq b \ right \}.}

Let a mapping be given in polar coordinates for which holds ${\ displaystyle \ Phi \ colon R \ to R, (r, \ theta) \ to (\ phi (r, \ theta), \ psi (r, \ theta))}$

${\ displaystyle \ Phi}$ is true to area,
${\ displaystyle \ phi (a, \ theta) = a, \ phi (b, \ theta) = b}$ for all , ${\ displaystyle \ theta}$
${\ displaystyle \ psi (a, \ theta) <\ theta, \ psi (b, \ theta)> \ theta}$ for everyone . ${\ displaystyle \ theta}$

Then has at least two fixed points. ${\ displaystyle \ Phi}$

literature

Henri Poincaré : Sur un théoréme de géométrie , Rend. Circ. Mat. Palermo 33, 374-407 (1912)
George David Birkhoff : An extension of Poincaré's last geometric theorem , Acta Math. 47, 297-311 (1925)
Morton Brown , Walter David Neumann : Proof of the Poincaré-Birkhoff fixed point theorem , Michigan Math. J. 24, 21–31 (1977)
Elmar Winkelnkemper : A generalization of the Poincaré-Birkhoff theorem , Proc. AMS 102, 1028-1030 (1988)

Individual evidence

↑ Guido Walz (Ed.), Lexikon der Mathematik, Volume 4, Springer Spectrum 2017, p. 218, ISBN 978-3-662-53499-1

[1] Guido Walz (Ed.), Lexikon der Mathematik, Volume 4, Springer Spectrum 2017, p. 218, ISBN 978-3-662-53499-1