Mathematics

he word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It’s been known since work of Ferenczi that linear word complexity highly constrains the dynamical behaviour of a subshift.

TBA

We focus on shift of finite type (SFTs) obtained by forbidding one word from an ambient SFT. Given a word in a one-dimensional full shift, Guibas and Odlyzko, and Lind showed that its auto-correlation, zeta function and entropy (of the corresponding SFT) all determine the same information. We first prove that when two words both have trivial auto-correlation, the corresponding SFTs not only have the same zeta function, but indeed are conjugate to each other, and this conjugacy is given by a chain of swap conjugacies. We extend this result to the case when the ambient SFT is the golden mean shift, with an additional assumption on the extender set of the words. Then, we introduce SFTs obtained by forbidding one finite pattern from a higher dimensional ambient SFT. We prove that, when two patterns have the same auto-correlation, then there is a bijection between the language of their corresponding SFTs. The proof is a simple combination of a replacement idea and the inclusion-exclusion principle. This is joint work with Nishant Chandgotia, Brian Marcus and Jacob Richey.

Krieger's celebrated embedding theorem gives necessary and sufficient conditions, in terms of periodic points, for proper embedding of a subshift into a mixing shift of finite type (SFT). The result does not generalize to mixing sofic shifts. Boyle introduced the notion of a receptive periodic point, which is the main obstruction. He also proved that a condition involving receptive periodic points is sufficient for proper embedding of a subshift into a mixing sofic shift. We introduce a stronger notion of embedding, called factorizable embedding, which requires the embedding to factor through a mixing SFT. We show that Boyle's sufficient condition for proper embedding into a mixing sofic shift is necessary and sufficient for a factorizable embedding into a mixing sofic shift. We also give a new characterization of receptive periodic points, and we give a characterization of mixing for irreducible sofic shifts in terms of receptive periodic points. This is joint work with Tom Meyerovitch, Klaus Thomsen and Chengyu Wu.