270 research outputs found
On closed subgroups of the group of homeomorphisms of a manifold
Let be a triangulable compact manifold. We prove that, among closed
subgroups of \homeo_{0}(M) (the identity component of the group of
homeomorphisms of ), the subgroup consisting of volume preserving elements
is maximal
An index for Brouwer homeomorphisms and homotopy Brouwer theory
We use the homotopy Brouwer theory of Handel to define a Poincar{\'e} index
between two orbits for an orientation preserving fixed point free homeomorphism
of the plane. Furthermore, we prove that this index is almost additive.Comment: To appear in Ergodic Theory and Dynamical System
Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique
Mary Rees has constructed a minimal homeomorphism of the 2-torus with
positive topological entropy. This homeomorphism f is obtained by enriching the
dynamics of an irrational rotation R. We improve Rees construction, allowing to
start with any homeomorphism R instead of an irrational rotation and to control
precisely the measurable dynamics of f. This yields in particular the following
result: Any compact manifold of dimension d>1 which carries a minimal uniquely
ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism
with positive topological entropy. More generally, given some homeomorphism R
of a (compact) manifold and some homeomorphism h of a Cantor set, we construct
a homeomorphism f which "looks like" R from the topological viewpoint and
"looks like" R*h from the measurable viewpoint. This construction can be seen
as a partial answer to the following realisability question: which measurable
dynamical systems are represented by homeomorphisms on manifolds
An introduction to Handel's homotopy Brouwer theory
Homotopy Brouwer theory is a tool to study the dynamics of surface
homeomorphisms. We introduce and illustrate the main objects of homotopy
Brouwer theory, and provide a proof of Handel's fixed point theorem. These are
the notes of a mini-course held during the workshop "Superficies en Montevideo"
in March 2012
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