Coassociative grammar, periodic orbits, and quantum random walk over ℤ

Abstract

Inspired by a work of Joni and Rota, we show that the combinatorics generated by a quantisation of the Bernoulli random walk over ℤ can be described from a coassociative coalgebra. Relationships between this coalgebra and the set of periodic orbits of the classical chaotic system x↦2x mod⁡1, x∈[0,1], are also given.