Stabilization for Iwasawa modules in Z_p-extensions

Let K/k be a Z_p-extension of a number field k with layers k_n. Let i_{n,m} be the map induced by inclusion between the p-parts of the class groups of k_n and k_m (m>n). We study the capitulation kernels H_{n,m}:=ker(i_{n,m}) and H_n:=\cup_{m>n} H_{n,m} to give some explicit formulas for their size and prove stabilization properties for their orders and p-ranks. We also briefly investigate stabilization properties for the cokernel of i_{m,n} and for the kernels of the norm maps and point out their relations with the nullity of the Iwasawa invariants for K/k.