On the sphere conjecture of Birkhoff

Abstract:

Birkhoff’s sphere conjecture, now known to be false, says that if $f$ is a measure preserving homeomorphism of ${S^2}$ with the poles $N$ and $S$ fixed, and with no other periodic points, then $f$ is topologically conjugate to an irrational rotation of ${S^2}$. In this setting we say that $D \subset {S^2}$ is maximal for $f$ if $f\left ( D \right ) \cap D = \emptyset$ and $D$ is maximal with respect to that property. Also, $f$ is $2$-small if for any circular ball $B$ such that $f\left ( B \right ) \cap B = \emptyset ,{f^{ - 1}}\left ( B \right ) \cap f\left ( B \right ) = \emptyset$ also. Theorem. Any $f$ as above and also $2$-small has a maximal set $D$ which is an open ball; its boundary contains $N$ and $S$ and is locally connected, and the area of $D$ is an irrational fraction of the area of ${S^2}$. This theorem gives another way of looking at the maps involved in the Birkhoff conjecture.