TITLE: Dynamics of Riemann zeta function
QUESTION [8 upvotes]: Has the dynamics of the Riemann zeta function been studied? By dynamics I mean the limiting behavior of the sequence of iterates $s, \zeta (s), \zeta (\zeta (s)), \zeta (\zeta (\zeta (s)))\dots $ for various starting values of $s$ in the (extended) complex plane. In particular, for which $s \in \mathbb{C}$ do the iterates $\zeta^{(n)}(s)$ converge to 0?

REPLY [3 votes]: There is also a fixed point near $s=1.83377$.  The Mathematica graphic below shows the argument of $\zeta(s)-s$, for $-10\le\sigma\le 5$, $-30\le t\le 30$,interpreted as a color.  We see the two real fixed points and the pole at $s=1$, as well as a number of complex conjugate pairs of fixed points.