Keith Burns

Manifolds of nonpositive curvature and their buildings

AbstractLet M be a complete Riemannian manifold of bounded nonpositive sectional curvature and finite volume. We construct a topological Tits building Δ

Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems

\tilde M

On topological tits buildings and their classification

associated to the universal cover of M. If M is irreducible and rank (M)≥2, we show that Δ

Stable ergodicity of skew products

\tilde M

Stable ergodicity and Anosov flows

is a building canonically associated with a Lie group and hence that M is locally symmetric.

Hyperbolic behaviour of geodesic flows on manifolds with no focal points

We prove that a diffeomorphism possessing a homoclinic point with a topological crossing (possibly with infinite order contact) has positive topological entropy, along with an analogous statement for heteroclinic points. We apply these results to study area-preserving perturbations of area-preserving surface diffeomorphisms possessing homoclinic and double heteroclinic connections. In the heteroclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the double heteroclinic connection or if it creates a homoclinic connection. In the homoclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the connection. These results significantly simplify the application of the Poincaré-Arnold-Melnikov-Sotomayor method. The results apply even when the contraction and expansion at the fixed point is subexponential.