Minimal entire functions

Consider the space of nonconstant entire functions ε with the topology of uniform convergence on compact subsets of ℂ and with the action of ℂ by translation. A minimal entire function is a nonconstant entire function f with the property that for any g ∈ ε which is a limit of translates of f, in turn, f is a limit of translates of g. Thus, X, the orbit closure of f is a minimal closed invariant set. It is not clear a priori that there exist such functions with X including functions that are not translates of f. I will show that many such functions can be constructed and that their orbit closures can be quite large and interesting from a dynamical point of view. The main example is based on the construction of a particular compact minimal action of ℝ² with rather special properties.