An exponentional iteration with complex connections

Extract

We have both used the iteration x₀ = 1, x_n+1 = c^xn (n ≥ 0) with groups of students to investigate the range of values of c > 0 for which the iterative schema converges. It is well known that there are real numbers p, q with 0 < p < q which divide the domain of positive reals into three intervals in which three distinct kinds of behaviour are exhibited. For values of c in T = (0, p) the iteration settles down to oscillations between two real values of x. For values of c in S = (p, q) the iteration converges to a single real value of x, and for values of c in D = (q, ∞) the iteration diverges. Approximations to p and q can be found by numerical investigations with calculators and computers, and the derivation of analytic expressions for p and q is within reach of many sixth-formers.