Algebraic dynamics and transcendental numbers

A first example of a connection between transcendental numbers and complex dynamics is the following. Let p and q be polynomials with complex coefficients of the same degree. A classical result of Bottcher states that p and q are locally conjugates in a neighborhood of oo: 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. A consequence 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. A quite different interplay between diophantine approximation and algebraic dynamics arises from the interpretation of the height of algebraic numbers in terms of the entropy of algebraic dynamical systems. Finally we say a few words on the work of J.H. Silverman on diophantine geometry and canonical heights including arithmetic properties of the Henon map.