An algorithm for periodicity and almost periodicity of uniform substitution sequences and its implications on aperiodic spin models

Substitution sequences are sequences generated by successive applications of deterministic substitution rules on n letters that are, in general, neither regular nor disordered. We propose a simple algorithm for classifying uniform substitutions sequences concerning periodicity and/or almost periodicity, based only on the substitution rules, whose implementation was recently registered. That classification is based on two theorems - that are also proved - about almost periodicity (extended for non-uniform substitutions) and periodicity, respectively. It recognizes automatically the character of the sequences concerning the periodicity and almost periodicity: purely periodic, ultimately periodic, weakly non periodic, and strongly non-periodic. That recognization selects the non-periodic sequences: among them, it also selects automatically the ones that promote relevance of geometrical fluctuations on the critical behavior of spin. models in relation to the homogeneous system. (C) 2019 Elsevier B.V. All rights reserved.