Multiplicative structures of hypercyclic functions for convolution operators

In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given entire function of subexponential type. A certain stability under multiplication is also shown for compositional hypercyclicity on complex domains.Comment: 12 page

Connectedness properties of the set where the iterates of an entire function are bounded

We investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. In particular, we describe a class of transcendental entire functions for which an analogue of the Branner-Hubbard conjecture holds and show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples to show that K(f) can be totally disconnected, and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.Comment: 21 page

On the Hausdorff dimension of the Julia set of a regularly growing entire function

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.Comment: 21 page