Coassociative Grammar, Periodic Orbits and Quantum Random Walk over Z 1

This work will be devoted to the quantisation of the classical Bernoulli random walk over Z. As this random walk is isomorphic to the classical chaotic dynamical system x 7→ 2x mod 1 with x ∈ [0, 1], we will explore the rôle of classical periodic orbits of this chaotic map in relation with a non commutative algebra associated with the quantisation of the Bernoulli walk. In particular we show that the set of periodic orbits, PO, of the map x 7→ 2x mod 1 can be embeded into a language equipped with a coassociative grammar and for any fixed time, that any vertex of Z is in one to one with of a subset of PO. The reading and the contraction maps applied to these periodic orbits allow us to recover the combinatorics generated by the quantum random walk over Z