Geometry, Topology and Dynamics Seminar
Olga Lukina
UIC
The dynamical classification of weak solenoids
Abstract: A matchbox manifold is a continuum which is locally homeomorphic to the product of R^n times Cantor set. In the case when n=1
and the matchbox manifold admits a flow, Aarts and Martens showed that two such continua are homeomorphic if and only they are
return equivalent, which means that there are Poincare maps associated to local sections in the continua whose actions are
conjugate. We show that such dynamical classification generalizes to certain classes of higher dimensional matchbox manifolds.
Namely, for n>1 two matchbox manifolds are return equivalent if their holonomy pseudogroups associated to local sections, are
isomorphic. A matchbox manifold is called Y-like, if it admits projections to Y with arbitrarily small fibres. Our first
result proves that two equicontinuous T^n-like matchbox manifolds are return equivalent if and only if they are homeomorphic.
Our second result requires Y to be an aspherical Borel manifold, whose every finite cover is also aspherical Borel. Then two
such equicontinuous Y-like matchbox manifolds containing a simply connected leaf are return equivalent if and only if they
are homeomorphic. Joint work with Alex Clark and Steve Hurder.
Monday January 27, 2014 at 3:00 PM in SEO 636