The Teichmuller Space of an Entire Function

We consider the Teichmuller space of a general entire transcendental function f : C → C regardless of the nature of the set of singular values of f, i.e., critical and asymptotic values. We prove that, as in the known case of periodic points and critical values, asymptotic values are also fixed points of any quasiconformal automorphism that commutes with f and which is homotopic to the identity, rel. the ideal boundary of the domain.

As a consequence, the general framework of McMullen and Sullivan [McMS98] for rational functions applies also to entire functions and we can apply it to study the Teichmuller space of f, analyzing each type of Fatou component separately. Baker domains were already considered in [FH06], but the consideration of wandering domains are new.

We provide different examples of wandering domains; each of them adding a different quantity to the dimension of the Teichmuller space. In particular, we give examples of rigid wandering domains.