TRANSCENDENCY OF LOCAL CONJUGACIES IN COMPLEX DYNAMICS AND TRANSCENDENCY OF THEIR VALUES

Let p and q be polynomials of the same degree. A classical result of B ottcher says that there exists a function f conformal in a neighborhoo d of infinity such that f(p(z)) = q(f(z)). We show that f is transcend ental and takes transcendental values at algebraic points unless p and q 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.