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Algebraic Dynamics and Transcendental Numbers

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Book cover Noise, Oscillators and Algebraic Randomness

Part of the book series: Lecture Notes in Physics ((LNP,volume 550))

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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 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. 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.

Aquite 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.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, Université P. et M. Curie, Théorie des Nombres, Case 247, 4, Place Jussieu, F-75252, PARIS Cedex 05, France

    Michel Waldschmidt

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  1. Michel Waldschmidt
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Editors and Affiliations

  1. Laboratoire de Physique et Merologie, des Oscillateurs du CNRS, 32 Avenue de l’Observatoire, 25044, Besançon Cedex, France

    Michel Planat

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Waldschmidt, M. (2000). Algebraic Dynamics and Transcendental Numbers. In: Planat, M. (eds) Noise, Oscillators and Algebraic Randomness. Lecture Notes in Physics, vol 550. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45463-2_19

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  • DOI: https://doi.org/10.1007/3-540-45463-2_19

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67572-3

  • Online ISBN: 978-3-540-45463-2

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