Abstract：

In the space O(C,C) of entire functions, equipped with the open-compact topology, an element is called universal if its translation orbit is dense. It is hypercyclic w.r.t some translation operator if its orbit under this operator is dense. It is fully hypercyclic if it is simultaneously hypercyclic to all non-trivial translations in all directions.

Universal entire functions are transcendental, hence their Nevanlinna characteristic functions grows faster than O(log r). Dinh-Sibony asked what the slowest Nevanlinna growth of universal entire curves is. In a joint work with Dinh Tuan Huynh and Song-Yan Xie, we solved their question completely, by constructing universal entire curves in CP^n whose Nevanlinna growth is slower than any given transcendental entire function.

Bin Guo and Song-Yan Xie discovered the conflict between fully hypercyclicity and slow growth. They proved that if the growth is too slow then the hypercyclic directions in [0,2pi) has Hausdorff dimension 0.

Replace C by the unit disc D, and translations by Aut(D), one can also talk about universal holomorphic discs. Transcendental functions defined on D with bounded Nevanlinna characteristic functions are called of bounded type, which is the analogous property of having slow growth. In a joint work with Bin Guo and Song-Yan Xie, we constructed universal discs in CP^n of bounded type. We also discovered a weak-conflict between fully hypercyclicity and slow growth. If the disc is of bounded type, then the hypercyclic directions in [0,2pi) has Lebesgue measure 0.