Two-cycles in the Infinite Exponential Tower

Abstract

The infinite exponential tower is studied through the associated iteration c₁ = 0 and cₙ₊₁ = eᶜₙ λ, for complex λ. For a subset of λ values, the sequence displays stable 2-cycles, that is to say as n → ∞ we observe that the odd subsequence c₂ₙ₋₁ → A whereas the even subsequence c₂ₙ → B, with A ≠ B. Thus, A and B obey B=eᴬ λ and A = eᴮ λ. Numerical investigations of the 2-cycles use a further transformation ζexp(-ζ) = λ = ln(z) and the set of ζ values corresponding to 2-cycles has a curious shape, reminding us of pictures of insect larva; the region has sharply scalloped edges. This paper gives an analytic expression for the edges of the 2-cycle region and a complete explanation of the cusps on the boundary that give the scalloped look.