Dynamics of surface homeomorphisms, topological versions of Leau-Fatou flower theorem and stable manifold theorem.
(Homéomorphismes de surfaces, théorèmes de la fleur de Leau-Fatou et de la variété stable.)

*(French)*Zbl 1073.37046
Astérisque 292. Paris: Société Mathématique de France (ISBN 2-85629-153-8/pbk). 120 p. (2004).

This paper studies local orientation-preserving homeomorphisms in the plane with a single fixed-point, or, equivalently by an extension argument, orientation-preserving homeomorphisms of the 2-sphere with exactly two fixed-points. The crucial dynamical distinction is the index of the fixed-point: if it is \(1+p\) with \(p>0\), then results similar to (but weaker than) the local dynamics of \(z\mapsto z+z^p\) are proved. If the index is \(1-p\) with \(p>0\), there is a cyclic collection of alternatingly stable and unstable ‘branches’, at least \(p\) many of each. This generalizes the invariant manifold theorem, which is the special case \(p=2\) in differentiable dynamics. However, in the topological category, these branches need not be (topological) manifolds, nor can there be a classification of dynamics away from these branches that would generalize Hartman-Grobman.

Before proving the theorems, the author explains by a wealth of examples how such limitations are unavoidable in the topological category. Informal abstract of proof ideas: A key tool is to construct a decomposition into ‘bricks’, following Sauzet, for an orientation-preserving homeomorphism of the 2-sphere with one fixed-point each of index \(1+p\) and \(1-p\), respectively. These bricks are such that neither intersects its image, but the union of any two adjacent ones does intersect its image. The brick decomposition gives rise to a coarse grained dynamics. From these bricks, Brouwer domains are constructed, each looking like the slice of an orange, with the dynamics inside each slice moving either north or south. Each of the local theorems arises from the theorem on the sphere.

Before proving the theorems, the author explains by a wealth of examples how such limitations are unavoidable in the topological category. Informal abstract of proof ideas: A key tool is to construct a decomposition into ‘bricks’, following Sauzet, for an orientation-preserving homeomorphism of the 2-sphere with one fixed-point each of index \(1+p\) and \(1-p\), respectively. These bricks are such that neither intersects its image, but the union of any two adjacent ones does intersect its image. The brick decomposition gives rise to a coarse grained dynamics. From these bricks, Brouwer domains are constructed, each looking like the slice of an orange, with the dynamics inside each slice moving either north or south. Each of the local theorems arises from the theorem on the sphere.

Reviewer: Jochen Denzler (Knoxville)

##### MSC:

37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |

37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |