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Solly Coles

Publications and source records attributed to Solly Coles.

5 recordsLinked to original sources
arXiv2026

Stable covers of subshifts

Given a dynamical system, a characteristic measure is a Borel probability measure invariant under all of its automorphisms. Frisch and Tamuz asked if every symbolic system supports such a measure. Motivated by this problem, 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 stabilize. The first is in terms of entropy decay and the second in terms of periodic points. We show that the first type of stabilization gives a new characterization of the class of language stable shifts and demonstrates that there is a mechanism for producing a characteristic measures that relies only on entropy differences. For the second type of stabilization, we show that this defines a new class of subshifts, invariant under conjugacies, that have characteristic measures.

math.DS↗
arXiv2024

Vassiliev invariants and writhe for periodic orbits of Axiom A flows

We obtain asymptotics for the average value taken by a Vassiliev invariant on knots appearing as periodic orbits of an Axiom A flow on $S^3.$ The methods used also give asymptotics for the writhe of periodic orbits. Our results are analogous to those of G. Contreras for average linking numbers.

math.DS↗
arXiv2022

Helicity, linking and the distribution of null-homologous periodic orbits for Anosov flows

This paper concerns connections between dynamical systems, knots and helicity of vector fields. For a divergence-free vector field on a closed $3$-manifold that generates an Anosov flow, we show that the helicity of the vector field may be recovered as the limit of appropriately weighted averages of linking numbers of periodic orbits, regarded as knots. This complements a classical result of Arnold and Vogel that, when the manifold is a real homology $3$-sphere, the helicity may be obtained as the limit of the normalised linking numbers of typical pairs of long trajectories. We also obtain results on the asymptotic distribution of weighted averages of null-homologous periodic orbits.

math.DS↗
arXiv2018

How to hear the shape of a billiard table

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.

math.DS↗