Transcendency of local conjugacies in complex dynamics and transcendency of their values

Abstract

Letp andq be polynomials of the same degree. A classical result of Bottcher says that there exists a functionf conformal in a neighborhood of infinity such thatf(p(z))=q(f(z)). We show thatf is transcendental and takes transcendental values at algebraic points unlessp andq are linearly conjugate to monomials or Chebychev polynomials. As an application, we show that the conformal map from the exterior of the Mandelbrot set onto the exterior of the unit disk takes transcendental values at algebraic points. A second application is the solution of a transcendency problem posed by Golomb.