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

Abstract

$p$-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 ${\mathbb{P}}^1\setminus \left\{0,{\mu }_N,\infty \right\}$. 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.