# What is the simplest SFT on $\mathbb{Z}^2$ that has no periodic points?

An SFT (shift of finite type) is a set of maps to some finite alphabet that is defined by a finite number of disallowed finite words.

By simple I mean has a small alphabet and a small number of disallowed words.

QUESTION. What are some of the simplest SFTs on $\mathbb{Z}^2$ that have no periodic points?

• And what do you mean by "small"? – YCor Dec 18 '16 at 20:32
• The alphabet is finite, as is the list of forbidden words, so small just means small cardinality. I guess that I would settle for anything with a simple description, in whatever sense. – Vladimir Dec 18 '16 at 23:41
• The simplest I know are the Robinson aperiodic tiling <a href=maa.org/sites/default/files/pdf/upload_library/22/Polya/… > and the Cari-Kulik tiling (mentioned by Doug Lind below) – Anthony Quas Dec 19 '16 at 2:22

This provides a $\mathbb{Z}^2$ SFT whose alphabet has 11 symbols, and with forbidden words that are pairs of adjacent (vertically or horizontally) tiles whose edges do not match.