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

TL;DR

This paper investigates the dependence of $p$-adic cyclotomic multiple zeta values on Frobenius iterations, introducing a new harmonic action to improve their computation and understanding.

Contribution

It introduces a new pro-unipotent harmonic action and analyzes how Frobenius iteration affects $p$-adic cyclotomic multiple zeta values, advancing computational methods.

Findings

New relations between Frobenius iterations and zeta values

Enhanced computational techniques for $p$-adic cyclotomic multiple zeta values

Definition of a novel pro-unipotent harmonic action

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 $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$. 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.