Frequently hypercyclic entire curves into projective spaces with growth order 1 (Zhangchi CHEN - East China Normal University)

An entire curve h : C → P^1(C) is called frequently hypercyclic with respect
to some translation T_a for some a ∈ C^∗, if for any non-empty open subset U ⊂ O(C,
Y ), the set of positive integers k ⩾ 1 with h(· + k a) ∈ U has positive lower density.
Frequent hypercyclicity was first discovered in the Riemann-zeta function,
acted by vertical translations. It motivates a lot of research in dynamics of linear
opeartors. Any frequently hypercyclic entire curve into projective spaces h : C → P^m has
growth order at least 1. Note that the Riemann-zeta function has growth order 1.
We construct order 1 entire curves h : C → P^m frequently hypercyclic with
respect to translations in countably many directions. This result on the slow growth
rate is optimal. This is a joint work with Bin Guo and Song-Yan Xie (arXiv:2409.08048).