Subshifts with slowly growing numbers of follower sets
Thomas French, Nic Ormes, Ronnie Pavlov · Contemporary mathematics - American Mathematical Society · 2016
For any subshift, define F X ( n ) F_X(n) to be the collection of distinct follower sets of words of length n n in X X . Based on a similar result proved in a work of Ormes and Pavlov, we conjecture that if there exists an n n for which | F X ( n ) | ≤ n |F_X(n)| \leq n , then X X is sofic. In this paper, we prove several results related to this conjecture, including verifying it for n ≤ 3 n \leq 3 , proving that the conjecture is true for a large class of coded subshifts, and showing that if there exists n n for which | F X ( n ) | ≤ log 2 ( n + 1 ) |F_X(n)| \leq \log _2(n+1) , then X X is sofic.