37 research outputs found
Transitive centralizers and fibered partially hyperbolic systems
We prove several rigidity results about the centralizer of a smooth
diffeomorphism, concentrating on two families of examples: diffeomorphisms with
transitive centralizer, and perturbations of isometric extensions of Anosov
diffeomorphisms of nilmanifolds.
We classify all smooth diffeomorphisms with transitive centralizer: they are
exactly the maps that preserve a principal fiber bundle structure, acting
minimally on the fibers and trivially on the base.
We also show that for any smooth, accessible isometric extension of an Anosov diffeomorphism of a nilmanifold, subject to a spectral
bunching condition, any sufficiently -close
to has centralizer a Lie group. If the dimension of this Lie group equals
the dimension of the fiber, then is a principal fiber bundle morphism
covering an Anosov diffeomorphism
Ergodic universality of some topological dynamical systems
The Krieger generator theorem says that every invertible ergodic measure-preserving system with finite measure-theoretic entropy can be embedded into a full shift with strictly greater topological entropy. We extend Krieger's theorem to include toral automorphisms and, more generally, any topological dynamical system on a compact metric space that satisfies almost weak specification, asymptotic entropy expansiveness, and the small boundary property. As a corollary, one obtains a complete solution to a natural generalization of an open problem in Halmos's 1956 book regarding an isomorphism invariant that he proposed