On Anosovity, divergence and bi-contact surgery

Abstract

We discuss a metric description of the divergence of a (projectively) Anosov flow in dimension 3, in terms of its associated growth rates and give metric and contact geometric characterizations of when a projectively Anosov flow is Anosov. Then, we study the symmetries that the existence of an invariant volume form yields on the geometry of an Anosov flow, from various viewpoints of the theory of contact hyperbolas, Reeb dynamics and Liouville geometry, and give characterizations of when an Anosov flow is volume preserving in terms of those theories. We finally use our study to show that the bi-contact surgery operations of Salmoiraghi can be applied in an arbitrary small neighborhood of a periodic orbit of any Anosov flow. In particular, we conclude that the Goodman surgery of Anosov flows can be performed using a bi-contact surgery of Salmoiraghi.

arXiv comment21 pages

References in corpus (10)

• Building Anosov flows on 3-manifolds
• Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
• On Invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture
• Dynamics and Topology of Conformally Anosov Contact 3-Manifolds
• Paires de structures de contact sur les variétés de dimension trois
• Approximation of Partially Smooth Functions
• Goodman surgery and projectively Anosov flows
• Ricci Curvature, Reeb Flows and Contact 3-Manifolds
• Goodman-Fried surgery, Birkhoff section, and R-covered Anosov flows
• Anosov flows on Dehn surgeries on the figure-eight knot

Cited by in corpus (4)

• On the existence of critical compatible metrics on contact $3$-manifolds
• Strongly adapted contact geometry of Anosov 3-flows
• Anosov flows and Liouville pairs in dimension three
• HyperKahler Contact Distributions