Research

My research mainly concerns hyperbolic flows, and the knot-theoretic properties of their periodic orbits. I’ve recently also been thinking about symbolic dynamics, and the effects of Ricci flow on the dynamics of geodesic flows. 

Publications

1. Vassiliev invariants and writhe for periodic orbits of Axiom A flows, Ergod. Theory Dyn. Syst 45, 2659-2671, 2025.
   We consider the average value taken by a Vassiliev invariant on the periodic orbits of an Axiom A flow on S^3. For periodic orbits of length approximately T, we show that the average grows polynomially in T (where the polynomial degree is explicitly given in terms of the invariant).

2. Distribution of periodic orbits in the homology group of a knot complement (joint with R. Sharp), Geom. Dedicata 218, 75, 2024.
   Consider a smooth Anosov flow on a three manifold M, with at least one null-homologous periodic orbit (e.g. a contact Anosov flow). Fixing such an orbit K, we count the other orbits according to their linking numbers with K (or more generally, their homology classes in M \ K).

3. Helicity, linking and the distribution of null-homologous periodic orbits for Anosov flows (joint with R. Sharp), Nonlinearity 36, 21-58, 2023.
   We consider the average linking number of periodic orbits of Anosov flows on three-manifolds. For periodic orbits of length approximately T, we show that the average is asymptotically CT^2, for an explicit constant C. When the flow is volume-preserving, the average linking number provides a characterisation of the hydrodynamical helicity, in terms of periodic orbits.
   
   

Preprints

1. Spectral gaps and mixing rates of primitive matrices (joint with R. Pavlov), arxiv:2611.10951, 2026.
   We obtain lower bounds on the spectral gap/upper bounds on the mixing rate for arbitrary finite-state (not necessarily reversible) primitive Markov chains. Our bounds are on the order of the conjectured worst-case examples, and give a significant improvement on the order obtained from classical arguments. Our improvements come from using the irreducibility distance and chainable index, rather than the primitivity index typically used. This also yields corresponding bounds for arbitrary nonnegative primitive matrices.

2. Stable covers of subshifts (joint with V. Cyr, B. Kra, R. Pavlov), arXiv:2602.04097, 2026.
   We study the natural cover of a subshift by its shift of finite type approximations, and two senses in which this cover can be said to stabilise. The first is in terms of entropy decay and the second in terms of periodic points. We show that the entropy stability gives a new characterisation of the class of language stable shifts, and that the periodic point stability gives a new class of shifts that have characteristic measures. 

3. How to hear the shape of a billiard table (joint with A. Calderon, D. Davis, J. Lanier, A. Oliveira), arXiv1806.09644, 2018.
   The bounce spectrum of a polygonal billiard table is the collection of all bi-infinite sequences of edge labels corresponding to billiard trajectories on the table. We give methods for reconstructing from the bounce spectrum of a polygonal billiard table both the cyclic ordering of its edge labels and the sizes of its angles. We also show that it is impossible to reconstruct the exact shape of a polygonal billiard table from any finite collection of finite words from its bounce spectrum. 

Thesis

On the distribution of periodic orbits and linking numbers for hyperbolic flows, University of Warwick, supervised by R. Sharp.