Nonminimal cyclic invariant subspaces of hyperbolic composition operators

Operators on function spaces acting by composition to the right with a fixed self-map ϕ are called composition operators. We denote them C_ϕ. Given ϕ, a hyperbolic disc automorphism, the composition operator C_ϕ on the Hilbert Hardy space H² is considered. The bilateral cyclic invariant subspaces K_f, f ∈ H², of C_ϕ are studied, given their connection with the invariant subspace problem, which is still open for Hilbert space operators. We prove that nonconstant inner functions u induce non–minimal cyclic subspaces K_u if they have unimodular, orbital, cluster points. Other results about K_u when u is inner are obtained. If f ∈ H² \ {0} has a bilateral orbit under C_ϕ, with Cesàro means satisfying certain boundedness conditions, we prove K_f is non–minimal invariant under C_ϕ. Other results proving the non–minimality of invariant subspaces of C_ϕ of type K_f when f is not an inner function are obtained as well.