Simplicity of homeo\((\mathbb{D}^2,\partial\mathbb{D}^2\), Area) and fragmentation of symplectic diffeomorphisms. (English) Zbl 1231.57030

Le Roux, Frédéric

J. Symplectic Geom. 8, No. 1, 73-93 (2010).

The paper is concerned with the simplicity of the group of homeomorphisms of the planar disc supported in the interior of the disc. The main result states that the simplicity of this group is equivalent to the following fragmentation property of the compactly supported homeomorphisms of the plane. This is called the Property (\(P_0\)):
There exists some positive integer \(m\) such that for any \(g\in \text{Homeo}_c(\mathbb R^2,\text{Area})\) of size less than one is the composition of at most \(m\) elements of \(\text{Homeo}_c(\mathbb R^2,\text{Area})\) of size less than \(1/2\).
Here, the size of a homeomorphism \(g\) is defined as the infimum of the area of a topological disc containing the support of \(g\). The author also investigates various fragmentations properties.

Reviewer: Jarek Kedra (Aberdeen)

MSC:

57S05	Topological properties of groups of homeomorphisms or diffeomorphisms
37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces

Keywords:

homeomorphisms of the dics; fragmentation property; simple group