Cusp algebras.
Any geometrically finite polynomial f of degree d ≥ 2 with connected Julia set is accessible by structurally stable sub-hyperbolic polynomials of the same degree. Moreover, they are topologically conjugate to f on their Julia sets.
Let Y be a Riemann surface with compact boundary embedded into a hyperbolic Riemann surface of finite type X. It is proved that the space of deformations D of Y into X is an open subset of the Teichmüller space T(X) of X. Furthermore, D has compact closure if and only if Y is simply connected or isomorphic to a punctured disk, and D= T(X) if and only if the components of X Y are all disks or punctured disks.