6 research outputs found
Weak mixing suspension flows over shifts of finite type are universal
Let S be an ergodic measure-preserving automorphism on a non-atomic
probability space, and let T be the time-one map of a topologically weak mixing
suspension flow over an irreducible subshift of finite type under a Holder
ceiling function. We show that if the measure-theoretic entropy of S is
strictly less than the topological entropy of T, then there exists an embedding
from the measure-preserving automorphism into the suspension flow. As a
corollary of this result and the symbolic dynamics for geodesic flows on
compact surfaces of negative curvature developed by Bowen and Ratner, we also
obtain an embedding from the measure-preserving automorphism into a geodesic
flow, whenever the measure-theoretic entropy of S is strictly less than the
topological entropy of the time-one map of the geodesic flow.Comment: 27 pages, 1 figur
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