An $S$-adic characterization of minimal subshifts with first difference of complexity $1 \leq p(n+1) - p(n) \leq 2$

TL;DR

This paper provides a new $S$-adic characterization of minimal subshifts with complexity difference between consecutive factors equal to 1 or 2, reducing the set of morphisms needed to five.

Contribution

It improves previous bounds by establishing an $S$-adic characterization with only five morphisms, solving the $S$-adic conjecture for this class of subshifts.

Findings

Reduced the set of morphisms to five for the characterization.

Solved the $S$-adic conjecture for subshifts with complexity difference 1 or 2.

Provided a more efficient description of minimal subshifts with low complexity.

Abstract

In [Ergodic Theory Dynam. System, 16 (1996) 663--682], S. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is $S$-adic with $\card S \leq 3^{27}$. In this paper, we improve this result by giving an $S$-adic charaterization of these subshifts with a set $S$ of 5 morphisms, solving by this way the $S$-adic conjecture for this particular case.