Todd Fisher

Hyperbolic Sets That are Not Locally Maximal

This papers addresses the following topics relating to the structure of hyperbolic sets: First, hyperbolic sets that are not contained in locally maximal hyperbolic sets. Second, the existence of a Markov partition for a hyperbolic set. We construct new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets. The first example is robust under perturbations and can be constructed on any compact manifold of dimension greater than one. The second example is robust, topologically transitive, and constructed on a 4-dimensional manifold. The third example is volume preserving and constructed on R 4 . We show that every hyperbolic set is included in a hyperbolic set with a Markov partition. Additionally, we describe a condition that ensures a hyperbolic set is included in a locally maximal hyperbolic set.

Trivial centralizers for axiom A diffeomorphisms

We show there is a residual set of non-Anosov C∞ axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. If M is a surface and 2 ≤ r ≤ ∞, then we will show there exists an open and dense set of Cr axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. Additionally, we examine commuting diffeomorphisms preserving a compact invariant set Λ where Λ is a hyperbolic chain recurrent class for one of the diffeomorphisms.

Trivial centralizers for codimension-one attractors

We show that if

Open sets of diffeomorphisms with trivial centralizer in the C1 topology

\Lambda

Topological properties of invariant sets for two-dimensional hyperbolic toral automorphisms

is a codimension-one hyperbolic attractor for a

The Topology of Hyperbolic Attractors on Compact Surfaces

C^r

Local perturbations of conservative C 1 diffeomorphisms

diffeomorphism

Blowup in a mass-conserving convection-diffusion equation with superquadratic nonlinearity

f

Entropy-expansiveness for partially hyperbolic diffeomorphisms

, where

Entropic stability beyond partial hyperbolicity

2\leq r\leq \infty