On the Hilbert $2$-class field towers of some cyclotomic ${\mathbb{Z}}_2$-extensions

In this paper, we study the length of the $2$-class field towers and the structure of the Galois groups $\mathrm{G}\mathrm{a}\mathrm{l}(\mathcal{L}(K_n)\mathrm{/}{K}_n)$ of the maximal unramified $2$-extensions of the layers $K_n$ of the cyclotomic ${\mathbb{Z}}_2$-extension of some special Dirichlet fields. The capitulation problem is investigated too.