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 Böttcher 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. 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 Hénon map. 1 Transcendental Values of Böttcher Functions For any complex number c ∈ C, define the polynomial pc ∈ C[z] by pc(z) = z + c. For n ≥ 1, let pc be the n-th iterate of pc: p1c(z) = pc(z) = z 2 + c, p2c(z) = pc(z 2 + c) = (z + c) + c, pc (z) = p n−1 c (z 2 + c) (n ≥ 2). The Mandelbrot set M can be defined as M = { c ∈ C | pc (0) does not tend to ∞ as n → ∞}. In 1982, A. Douady and J. Hubbard have shown that M is connected. They constructed a conformal map Φ : C \M −→ {z ∈ C ; |z| > 1} from the complement of M onto the exterior of the unit disk, which is defined as follows. 2 Michel Waldschmidt For each c ∈ C, there is a unique power series φc with coefficients in Q(c), φc(z) = z + c0 + c1 z + c2 z + . . . ∈ Q(c)((1/z)), such that φc(z 2 + c) = φc(z) . For c 6∈ M , φc defines an analytic function near c. Then the above mentioned map Φ is defined by Φ(c) = φc(c). According to P.G. Becker, W. Bergweiler and K. Nishioka [2], [3], [11], for any algebraic α ∈ C \M , the number Φ(α) is transcendental. . The function φc is the unique Böttcher function with respect to pc = z +c and z. More generally, let p = az + · · · and q = bz + · · · be two polynomials in C[z] of degree d ≥ 2 and let λ ∈ C satisfy λ = a/b. There exists a unique function f , which is defined and meromorphic in a neighborhood of ∞, such that lim z→∞ f(z) λz = 1 and f (