The structure of low complexity subshifts

Abstract

An idea that became unavoidable to study zero entropy symbolic dynamics is that the dynamical properties of a system induce in it a combinatorial structure. An old problem addressing this intuition is finding a structure theorem for linear-growth complexity subshifts using the S-adic formalism. It is known as the S-adic conjecture and motivated several influential results in the theory. In this article, we completely solve the conjecture by providing an S-adic structure for this class. Our methods extend to nonsuperlinear-complexity subshifts. An important consequence of our main results is that these complexity classes gain access to the S-adic machinery. We show how this provides a unified framework and simplified proofs of several known results, including the pioneering 1996 Cassaigne's Theorem.