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

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 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|≤ 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.

READ FULL TEXT