Substitution systems and nonextensive statistics

Substitution systems evolve in time by generating sequences of symbols from a finite alphabet: At a certain iteration step, the existing symbols are systematically replaced by blocks of N-k symbols also within the alphabet (with N-k, a natural number, being the length of the kth block of the substitution). The dynamics of these systems leads naturally to fractals and self-similarity. By using 2-calculus (Garcia-Morales, 2012) universal maps for deterministic substitution systems both of constant and non-constant length, are formulated in 1D. It is then shown how these systems can be put in direct correspondence with Tsallis entropy. A 'Second Law of Thermodynamics' is also proved for these systems in the asymptotic limit of large words. (C) 2015 Elsevier B.V. All rights reserved.