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In [Ergodic Theory Dynam. System, 16 (1996) 663--682], S. Ferenczi provedthat 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 bygiving an $S$-adic charaterization of these subshifts with a set $S$ of 5morphisms, solving by this way the $S$-adic conjecture for this particularcase.

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