A systematic approach to computing and indexing the fixed points of an iterated exponential

Abstract

This paper describes a systematic method of numerically computing and indexing fixed points of $z^{z^w}$ for fixed $z$ or equivalently, the roots of $T_2(w;z)=w-z^{z^w}$. The roots are computed using a modified version of fixed-point iteration and indexed by integer triplets $\{n,m,p\}$ which associate a root to a unique branch of $T_2$. This naming convention is proposed sufficient to enumerate all roots of the function with $(n,m)$ enumerated by $\mathbb{Z}^2$. However, branches near the origin can have multiple roots. These cases are identified by the third parameter $p$. This work was done with rational or symbolic values of $z$ enabling arbitrary precision arithmetic. A selection of roots up to order $\{10^{12},10^{12},p\}$ with $|z|\leq 10^{12}$ was used as test cases. Results were accurate to the precision used in the computations, generally between $30$ and $100$ digits. Mathematica ver. $12$ was used to implement the algorithms.