Pro-unipotent harmonic actions and dynamical properties of p-adic cyclotomic multiple zeta values

p-adic cyclotomic multiple zeta values depend on the choice of a number of iterations of the crystalline Frobenius of the pro-unipotent fundamental groupoid of P-1 \ {0, mu(N), infinity}. In this paper we study how the iterated Frobenius depends on the number of iterations, in relation with the computation of p-adic cyclotomic multiple zeta values in terms of cyclotomic multiple harmonic sums. This provides new results on that computation and the definition of a new pro unipotent harmonic action.