Continued Fraction Algorithms, Functional Operators, and Structure Constants Continued Fraction Algorithms, Functional Operators, and Structure Constants 1 Continued Fraction Algorithms, Functional Operators, and Structure Constants

Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2-dimensional generalization. This paper surveys the main properties of functional operators, |transfer operators| due to Ruelle and Mayer (also following L evy, Kuzmin, Wirsing, Hensley, and others) that describe precisely the dynamics of the continued fraction transformation. Spectral characteristics of transfer operators are shown to have many consequences, like the normal law for logarithms of continuants associated to the basic continued fraction algorithm, where better convergence terms are obtained , and a purely analytic estimation of the average number of steps of the Euclidean algorithm. Transfer operators also lead to a complete analysis of the \Hakmem" algorithm for comparing two rational numbers via partial continued fraction expansions, and of the \digital tree" algorithm for completely sorting n real numbers by means of their continued fraction representations. Thus, a small number of \structure constants" appear to govern the behaviour of a variety of continued fraction based algorithms. Algorithmes de fractions continues, op erateurs fonctionnels et constantes de structure R esum e. Les fractions continues sont au ccur de nombreux algorithmes, comme l'algorithme d'Euclide ou l'algorithme de r eduction des r eseaux de Gauss qui en constitue une g en eralisation en dimension 2. Nous passons en revue les principales propri et es d'op erateurs fonctionnels |op erateurs de transfer| dus a Ruelle et Mayer (faisant suite a L evy, Kuzmin, Wirsing, Hensley et d'autres), qui d ecrivent pr ecis ement la dynamique de la transformation des fractions continues. Les propri et es spectrales de ces op erateurs ont de nombreuses cons equences, comme: la loi normale des logarithmes de continuants o u sont obtenus de meilleurs termes d'erreur, ainsi qu'une estimation purement analy-tique du nombre moyen d' etapes de l'algorithme d'Euclide. Les op erateurs de transfert conduisent de surcro^ t a une analyse compl ete de l'algorithme \Hak-mem" de comparaison de rationnels et de l'algorithme de tri digital appliqu e aux fractions continues. Il appara^ t ainsi qu'un petit nombre de constantes de structure gouvernent le comportement d'une vari et e d'algorithmes fond es sur les fractions continues Abstract Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2-dimensional generalization. This paper surveys the …