A $Z^k$-invariant subspace without the wandering property

We study operators of multiplication by $z^k$ in Dirichlet-type spaces $D_{\alpha }$. We establish the existence of $k$ and $\alpha$ for which some $z^k$-invariant subspaces of $D_{\alpha }$ do not satisfy the wandering property. As a consequence of the proof, any Dirichlet-type space accepts an equivalent norm under which the wandering property fails for some space for the operator of multiplication by $z^k$, for any $k\ge 6$.