Genericity of wild holomorphic functions and common hypercyclic vectors

Abstract

Let T α be the translation operator by α in the space of entire functions H( C) defined by T α( f )(z)=f(z+α) . We prove that there is a residual set G of entire functions such that for every f∈ G and every α∈ C⧹{0} the sequence T α n( f ) is dense in H( C) , that is, G is a residual set of common hypercyclic vectors ( functions) for the family {T α : α∈ C⧹{0}} . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.