Superexponentiation and Fixed Points of Exponential and Logarithmic Functions

where exponentiation occurs n times. Superexponentiation simply continues the pattern of addition, multiplication, and exponentiation. It seems to have first appeared in the literature in [4], for the purpose of exhibiting extremely large (albeit finite) numbers (its implicit use in [7], an earlier work than [4], is shown in [5]). Superexponentiation is used in [5] and [6] to examine the logical foundation of mathematical induction. In [2] superexponentiation and its inverse operation, iteration of logarithms, are used to analyze the running time of certain algorithms. A general discussion of superexponentiation is given in [1]. For our purposes, superexponentiation naturally arises from analyzing fixed points of exponential functions (and hence of the corresponding logarithms). We approach this using orbit analysis as in [3], i.e., by iterating the exponential function F(x) = b, starting with input x = b. For b > 1 the resulting orbit clearly will be strictly increasing, and for bases such as 2 or 10 (those used in the applications cited above), the orbit will diverge rapidly to infinity. Will this divergence occur for all b > 1? What kind of behavior is observed if 0 < b < 1? Will the orbit converge to a single number for suitable values of b? In Section 2 we discuss the context under which the author arrived at the problem. In Section 3 we answer the question of convergence, finding the set of b for which the orbit does indeed converge; this result is stated as a theorem at the end of the section. In Section 4 we offer several suggestions for further exploration.