Thermodynamic Formalism and Homological Characteristics of Anosov Flows

This project is on the interface between ergodic theory: the branch of the theory of dynamical systems that uses mathematical analysis to understand statistical properties, and low-dimensional topology. The aim of this project is to use techniques from thermodynamic formalism to study the distribution of periodic orbits of flows subject to homological constraints, and find asymptotic results for related quantities, such as the linking numbers of periodic orbits.
A starting point for the project is the work of Contreras, who used the equidistribution theory of Bowen to evaluate asymptotic average linking numbers of periodic orbits of hyperbolic flows in the 3-sphere. It seems plausible that one could utilise the more general equidistribution theory that now exists through thermodynamic formalism, to generalise Contreras' work to weighted averages and arbitrary 3-manifolds. Following this, one would hope to be able to make a connection between average linking numbers and hydrodynamical helicity; the helicity is already known to have a form similar to these average linking numbers, due to the work of Arnold and Vogel. This will be the main aim of the project.
There are many pre-existing results on how periodic orbits are distributed in homology classes. Any further work in this project will be an attempt to carry these results over to the context of ramified homology classes, where instead of the manifold's first homology group, one looks at the first homology group of the complement of a periodic orbit.
The research is in the areas of geometry, topology and Mathematical analysis and is wholly writhing the Mathematical Sciences themer.