Number Theory – Dynamics of the Riemann Zeta Function

Has the dynamics of the Riemann zeta function been studied? By dynamics, I understand the limiting behavior of the sequence of iterates $$s, zeta (s), zeta ( zeta (s)), zeta ( zeta ( zeta (s))) Points$$ for different starting values ​​of $$s$$ in the complex (extended) plane. In particular, for which $$s in mathbb {C}$$ do the iterations $$zeta ^ {(n)} (s)$$ to converge at 0?