On the distribution of periodic orbits and linking numbers for hyperbolic flows

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Abstract

This thesis concerns certain knot theoretic properties of the periodic orbits of hyperbolic flows, with a focus on Anosov flows.

Anosov flows are a class of hyperbolic dynamical system which were introduced in [Ano67] to generalise the behaviour of geodesic flows over negatively curved spaces. They have a countable infinity of periodic orbits which, when the flow is on a 3-dimensional manifold, can be studied as knots. We will study the linking numbers of these orbits with one another.

Our first result relates the helicity of certain Anosov flows to a weighted average of linking numbers of their periodic orbits. This complements a classical result of Arnol’d and Vogel ([Arn86], [Vog02]) which is that, when the manifold is a real homology 3-sphere, the helicity of a volume-preserving flow may be obtained as the limit of normalised linking numbers of long trajectories. Our result is also inspired by work of Contreras [Con95], who studied average linking numbers of hyperbolic flows in S3.

We then study the number and distribution of periodic orbits with prescribed linking properties relative to a fixed set of orbits. This is inspired by work of McMullen [McM13], which we use to present a new application of the methods of Babillot-Ledrappier [BL98], for counting orbits subject to certain constraints.

The methods used to prove these results come largely from thermodynamic formalism, along with the symbolic coding procedure for hyperbolic flows, developed independently by Bowen [Bow73] and Ratner [Rat73].

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