Substitution systems and nonextensive statistics

Abstract

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 Nk symbols also within the alphabet (with Nk, 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 B-calculus (García-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.