Dynamics of surface homeomorphisms Topological versions of the   Leau-Fatou flower theorem and the stable manifold theorem
  The study of the dynamics of a surface homeomorphism in the neighbourhood of an isolated fixed point leads us to the following results. If the fixed point index is greater than 1, a family of attractive and repulsive petals is constructed, generalizing the Leau-Fatou flower theorem in complex dynamics. If the index is less than 1, we get a family of stable and unstable branches, generalizing the stable manifold theorem in hyperbolic dynamics. 
Frederic Le Roux