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Continued fractions, modular symbols, and noncommutative geometry
 Selecta Mathematica (New Series) Vol.8 N.3
, 2002
"... Abstract. Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to th ..."
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Cited by 56 (18 self)
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Abstract. Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient PGL(2,Z) \ P 1 (R) should be considered as non–commutative modular curve, and show that the modular complex can be seen as a sequence of K0–groups of the related crossed–product C ∗ –algebras. §0. Introduction and summary In this paper we study the interrelation between several topics: a generalization of the classical Gauss problem on the distribution of continued fractions, certain averages of modular symbols, the properties of geodesics on modular curves, the Mixmaster Universe model in general relativity, and the non–commutative geometry
An Averagecase Analysis of the Gaussian Algorithm for Lattice Reduction
, 1996
"... .The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, the probability distribution decays geometrically, and the dynamics is characterized by a ..."
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Cited by 47 (9 self)
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.The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, the probability distribution decays geometrically, and the dynamics is characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Analysis in the discrete model and detailed numerical data are also presented. Une analyse en moyenne de l'algorithme de Gauss de r'eduction des r'eseaux R'esum'e. L'algorithme de r'eduction des r'eseaux en dimension 2 qui est du `a Gauss est analys'e sous sa forme dite standard. Il est 'etabli ici que, sous un mod`ele continu, sa complexit'e est constante en moyenne et que la distribution de probabilit'es associ'ee decroit g'eom'etriquement tandis que la dynamique est caract'eris'ee par une densit'e conditionnelle invariante. Les preuves f...
Continued Fraction Algorithms, Functional Operators, and Structure Constants
, 1996
"... 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 2dimensional generalization. This paper surveys the main properties of functional operators,  transfer o ..."
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Cited by 31 (6 self)
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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 2dimensional generalization. This paper surveys the main properties of functional operators,  transfer operators  due to Ruelle and Mayer (also following Lévy, 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 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 ...
Dynamical Analysis of a Class of Euclidean Algorithms
"... We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properti ..."
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Cited by 19 (5 self)
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We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 16 (6 self)
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We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
Limiting modular symbols and the Lyapunov spectrum
 J. Number Theory
"... This paper consists of variations upon a theme, that of limiting modular symbols introduced in [16]. We show that, on level sets of the Lyapunov exponent for the shift map T of the continued fraction expansion, the limiting modular symbol can be computed as a Birkhoff average. We show that the limit ..."
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Cited by 15 (10 self)
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This paper consists of variations upon a theme, that of limiting modular symbols introduced in [16]. We show that, on level sets of the Lyapunov exponent for the shift map T of the continued fraction expansion, the limiting modular symbol can be computed as a Birkhoff average. We show that the limiting modular symbols vanish
Modular curves, C ∗ algebras, and chaotic cosmology, in “Frontiers in Number Theory, Physics and Geometry
 Department of Mathematics, California Institute of Technology
, 2007
"... We make some brief remarks on the relation of the mixmaster universe model of chaotic cosmology to the geometry of modular curves and to noncommutative geometry. We show that the full dynamics of the mixmaster universe is equivalent to the geodesic flow on the modular curve X Γ0(2). We then consider ..."
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Cited by 6 (3 self)
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We make some brief remarks on the relation of the mixmaster universe model of chaotic cosmology to the geometry of modular curves and to noncommutative geometry. We show that the full dynamics of the mixmaster universe is equivalent to the geodesic flow on the modular curve X Γ0(2). We then consider a special class of solutions, with bounded number of cycles in each Kasner era, and describe their dynamical properties (invariant density, Lyapunov exponent, topological pressure). We relate these properties to the noncommutative geometry of a moduli space of such solutions, which is given by a Cuntz–Krieger C ∗algebra. 1 Modular curves Let G be a finite index subgroup of Γ = PGL(2, Z), and let XG denote the quotient XG = G\H 2, where H 2 is the 2dimensional real hyperbolic plane, namely the upper half plane {z ∈ C: ℑz> 0} with the metric ds 2 = dz  2 /(ℑz) 2. Equivalently, we identify H 2 with the Poincaré disk {z: z  < 1} with the metric ds 2 = 4dz  2 /(1 − z  2) 2. Let P denote the coset space P = Γ/G. We can write the quotient XG equivalently as XG = Γ \(H 2 × P). The quotient space XG has the structure of a noncompact Riemann surface, which can be compactified by adding cusp points at infinity: ¯XG = G\(H 2 ∪ P 1 (Q)) ≃ Γ \ ( (H 2 ∪ P 1 (Q)) × P). (1) In particular, we consider the congruence subgroups G = Γ0(p), with p a prime, given by matrices () a b g = c d satisfying c ≡ 0 mod p. In fact, for our purposes, we are especially interested in the case p = 2.2 Matilde Marcolli 1.1 Shift operator and dynamics If we consider the boundary P 1 (R) of H 2, the arguments given in [9] and [10] show that the quotient Γ \(P 1 (R) × P), better interpreted as a “noncommutative space”, gives rise to a compactification of the modular curve XG with a structure richer than the ordinary algebrogeometric compactification by cusp points. In [9] this was described in terms of the following dynamical system, generalizing the classical Gauss shift of the continued fraction expansion: T(x, t) = T: [0, 1] × P → [0, 1] × P
Euclidean dynamics
 DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
, 2006
"... We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer ope ..."
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Cited by 4 (2 self)
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We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various tools of analytic combinatorics: generating functions, Dirichlet series, Tauberian theorems, Perron’s formula and quasipowers theorems. Such dynamical analyses can be used to perform the averagecase analysis of algorithms, but also (dynamical) analysis in distribution.
On The GaussKuzminWirsing Constant
, 1995
"... . We give strong supporting evidence for the fact that the GaussKuzminWirsing constant is j 2 j = 0:30366 30028 98732 6585974481 21901 55623 \Sigma 10 \Gamma31 : If x 0 is uniformly distributed in [0; 1], then the nth iterate x n = T (n) (x 0 ) of the continued fraction transformation T (x) ..."
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Cited by 2 (0 self)
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. We give strong supporting evidence for the fact that the GaussKuzminWirsing constant is j 2 j = 0:30366 30028 98732 6585974481 21901 55623 \Sigma 10 \Gamma31 : If x 0 is uniformly distributed in [0; 1], then the nth iterate x n = T (n) (x 0 ) of the continued fraction transformation T (x) = f 1 x g (where fxg = x \Gamma bxc denotes the fractional part) has a distribution F n (x) = Pr(x n ! x) that converges geometrically to the invariant density F1 (x) found by Gauß, namely, F1 (x) = log 2 (1 + x) : The rate of convergence is exponential being of the form O( n 2 ), as established by Wirsing [8] using operator methods (the PerronFrobenius theory). Babenko [1] and Mayer (see [7] for a survey) later provided a complete spectral decomposition of the form F n (x) = F1 (x) + 1 X k=2 n k \Psi k (x); for some eigenvalues k and some analytic eigenfunctions \Psi k (x). In the course of a joint work with H. Daud'e relative to an averagecase analysis of lattice reduction...
Marcolli M.: On the distribution of continued fractions and modular symbols, Preprint of MPIM
, 2001
"... Abstract. Using the techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of consecutive convergents. We then study some averages rel ..."
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Cited by 1 (0 self)
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Abstract. Using the techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of consecutive convergents. We then study some averages related to modular symbols. §0. Introduction and summary 0.1. Continued fractions. We start by fixing the notation which will be used throughout the paper. Considering first k1,..., kn as independent variables, put for n ≥ 1