Univalent mappings and invariant subspaces of the Bergman and Hardy spaces

In both the Bergman space $A^2$ and the Hardy space $H^2$, the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function $g$ in $H^{\infty}$ has the wandering property in $X$, where $X$ denotes either $A^2$ or $H^2$, provided that every $g$-invariant subspace $M$ of $X$ is generated by the orthocomplement of $gM$ within $M$. It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of $H^{\infty}$ also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.