On distribution formulas for complex and $\mathrm{\ell }$-adic polylogarithms

We study an $\mathrm{\ell }$-adic Galois analogue of the distribution formulas for polylogarithms with special emphasis on path dependency and arithmetic behaviors. As a goal, we obtain a notion of certain universal Kummer-Heisenberg measures that enable interpolating the $\mathrm{\ell }$-adic polylogarithmic distribution relations for all degrees.