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?