分野別セミナー

福岡複素解析セミナー
https://sites.google.com/view/scvfukuoka/

アブストラクト：
In the space $\mathcal{O}(\mathbb{C},\mathbb{P}^n)$ of entire curves with the open-compact topology, an element is called universal if its translation orbit is dense. It is hypercyclic under some translation operator if its orbit under this operator is dense. It is fully hypercyclic if it is simultaneously hypercyclic under all translations in all directions.
Universal entire curves are transcendental, hence their Nevanlinna characteristic functions grow faster than $O(\log r)$. Dinh-Sibony asked about the slowest growth. In a joint work with Dinh Tuan Huynh and Song-Yan Xie, we solved this problem by constructing universal entire curves whose Nevanlinna characteristic functions grow slower than any given transcendental entire curve.
Bin Guo and Song-Yan Xie discovered the conflict between full hypercyclicity and slow growth. They proved that if the growth is too slow then the hypercyclic directions in $[0,2\pi)$ has Hausdorff dimension $0$.
Replace $\mathbb{C}$ by the unit disc $\mathbb{D}$, and translations by fixed-point-free automorphisms of $\mathbb{D}$, one can talk about universal holomorphic discs. Transcendental functions defined on $\mathbb{D}$ 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 $\mathbb{P}^n$ of bounded type. We also discovered a weak-conflict between full hypercyclicity and slow growth. If the disc is of bounded type, then the hypercyclic directions in $[0,2\pi)$ has Lebesgue measure $0$.