Shifts of finite type defined by one forbidden block and a Theorem of Lind

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.