Topological behaviour of logarithmic invariants

Let $\mathrm{\ell }$ be a rational prime number and $K$ a number field. We prove that the logarithmic module $X_d$ attached to a ${\mathbb{Z}}_{\mathrm{\ell }}^d$-extension $K_d$ of $K$ is a noetherian ${\mathrm{\Lambda }}_d$-module. Moreover, under the Gross-Kuz’min conjecture we prove that it is also torsion. We exploit this fact to deduce local and global information of the logarithmic invariants $\tilde{\mu }$ and $\tilde{\lambda }$ of ${\mathbb{Z}}_{\mathrm{\ell }}$-extensions.