Algebraic Dynamics and Transcendental Numbers

Abstract

A first example of a connection between transcendental numbers and complex dynamics is the following. Let p and q be polynomials with complex coe efficients of the same degree. Aclassical result of Bottcher states that p and q are locally conjugates in a neighborhood of ∞: there exists a function f, conformal in a neighborhood of infinity, such that f(p(z)) = q(f(z)). Under suitable assumptions, f is a transcendental function which takes transcendental values at algebraic points. Aconsequence is that the conformal map (Douady-Hubbard) from the exterior of the Mandelbrot set onto the exterior of the unit disk takes transcendental values at algebraic points. The underlying transcendence method deals with the values of solutions of certain functional equations.