Limits of iterations of complex maps and hypergeometric functions

Abstract

We consider the limit of the iteration of a map z ↦ m(z) from a complex domain D to D. For two kinds of maps m, we show that each iteration mⁿ(z) of m(z) converges for any z ∈ D as n → ∞ and that this limit is expressed by the hypergeometric function. These are analogs of the expression of the arithmetic-geometric mean by the Gauss hypergeometric function.