Amie Wilkinson

Stable ergodicity of the time-one map of a geodesic flow

We prove that the time-one map of the geodesic flow for a closed, negatively curved surface is stably ergodic.

Absolutely Singular Dynamical Foliations

Abstract: Let A3 be the product of the automorphism of T2 and of the identity on T1. A small perturbation g of A3 among volume preserving diffeomorphisms will have an invariant family of smooth circles Γ forming a continuous foliation of T3. Corresponding to the vector bundle tangent to the circles Γ there is a “central” Lyapunov exponent of (g, volume), which is nonzero for an open set of ergodic gs. This surprising result of Shub and Wilkinson is complemented here by showing that the volume on T3 has atomic conditional measures on the Γs: there is a finite k such that almost every Γ carries

Stable ergodicity of skew products

k

Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows

atoms of mass 1/k.

Stable ergodicity and Anosov flows

Abstract Stable ergodicity is dense among compact Lie group extensions of Anosov diffeomorphisms of compact manifolds. Under the additional assumption that the base map acts on an infranilmanifold, an extension that is not stably ergodic must have a factor that has one of three special forms. A consequence is that stable ergodicity and stable ergodicity within skew products are equivalent in this case.

Diffeomorphisms with positive metric entropy

We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.

Global rigidity of solvable group actions on S 1

In this note we prove that if M is a 3-manifold and ϕt:M→M is a C2, volume-preserving Anosov flow, then the time-1 map ϕ1 is stably ergodic if and only if ϕt is not a suspension of an Anosov diffeomorphism.

Stably ergodic approximation: two examples

We obtain a dichotomy for C1

Conservative partially hyperbolic dynamics

C^{1}

Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms

-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.