Rotation Vectors for Homeomorphisms of Non-Positively Curved Manifolds

TL;DR

This paper generalizes the concept of rotation vectors from torus homeomorphisms to homeomorphisms of non-positively curved manifolds, establishing their existence and applications in dynamical systems.

Contribution

It extends rotation vector theory to broader geometric contexts and demonstrates their use in constructing semi-conjugacies with geodesic flows.

Findings

Rotation vectors exist for almost every orbit under invariant measures.

Generalization to flows broadens applicability.

Non-null rotation vectors facilitate semi-conjugacy construction.

Abstract

Rotation vectors, as defined for homeomorphisms of the torus that are isotopic to the identity, are generalized to such homeomorphisms of any complete Riemannian manifold with non-positive sectional curvature. These generalized rotation vectors are shown to exist for almost every orbit of such a dynamical system with respect to any invariant measure with compact support. The concept is then extended to flows and, as an application, it is shown how non-null rotation vectors can be used to construct a measurable semi-conjugacy between a given flow and the geodesic flow of a manifold.