Dynamics of a certain sequence of powers. (Q1586753)

Summary: For any nonzero complex number \(z\) we define a sequence \(a_1(z) = z\), \(a_2(z) = z^{a_1(z)}\), \(\dotsc\), \(a_{n+1}(z) = z^{{a_n}(z)}\), \(n \in \mathbb N\). We attempt to describe the set of these \(z\) for which the sequence \(\{a_n(z)\}\) is convergent. While it is almost impossible to characterize this convergence set in the complex plane \(\mathbb C\), we achieved it for positive reals. We also discussed some connection to the Euler's functional equation.