arXiv subjects

Sophie MacDonald

Publications and source records attributed to Sophie MacDonald.

4 recordsLinked to original sources

arXiv2023

The road problem and homomorphisms of directed graphs

We make progress on a generalization of the road (colouring) problem. The road problem was posed by Adler-Goodwyn-Weiss and solved by Trahtman. The generalization was posed, and solved in certain special cases, by Ashley-Marcus-Tuncel. We resolve two new families of cases, of which one generalizes the road problem and follows Trahtman's solution, and the other generalizes a result of Ashley-Marcus-Tuncel with a proof quite different from theirs. Along the way, we prove a universal property for the fiber product of certain graph homomorphisms, which may be of independent interest. We provide polynomial-time algorithms for relevant constructions and decision problems.

math.CO↗

arXiv2022

Encoding subshifts through sliding block codes

We prove a generalization of Krieger's embedding theorem, in the spirit of zero-error information theory. Specifically, given a mixing shift of finite type $X$, a mixing sofic shift $Y$, and a surjective sliding block code $π: X \to Y$, we give necessary and sufficient conditions for a subshift $Z$ of topological entropy strictly lower than that of $Y$ to admit an embedding $ψ: Z \to X$ such that $π\circ ψ$ is injective.

math.DS↗

arXiv2020

Conformal measures and the Dobrushin-Lanford-Ruelle equations

We demonstrate the equivalence of two definitions of a Gibbs measure on a subshift over a countable group, namely a conformal measure and a Gibbs measure in the sense of the Dobrushin-Lanford-Ruelle (DLR) equations. We formulate a more general version of the classical DLR equations with respect to a measurable cocycle, which reduce to the classical equations when the cocycle is induced by an interaction or a potential, and show that a measure satisfying these equations must be conformal. To ensure the consistency of these results with earlier work, we review methods of constructing an interaction from a potential and vice versa, such that the interaction and the potential constructed from it, or vice versa, induce the same cocycle.

math.DS↗

arXiv2020

A Dobrushin-Lanford-Ruelle theorem for irreducible sofic shifts

We show that for a potential with summable variations on an irreducible sofic shift in one dimension, the equilibrium measures are precisely the shift-invariant Gibbs measures. The main tool in the proof is a preservation of Gibbsianness result for almost invertible factor codes on irreducible shifts of finite type, which we then extend to finite-to-one codes by applying the results about equilibrium measures.

math.DS↗