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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
Holography principle and arithmetic of algebraic curves
"... Abstract. According to the holography principle (due to G. ‘t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating thi ..."
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Cited by 33 (9 self)
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Abstract. According to the holography principle (due to G. ‘t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS3 holography of black holes. Moreover, in the case of Euclidean AdS2 holography, we present some results on bulk/boundary correspondence where the boundary is a non–commutative space. 0.1. Holography principle. Consider a manifold M d+1 (“bulk space”) with boundary N d. The holography principle postulates the existence of strong ties between certain field theories on M and N respectively. For example, in the actively discussed Maldacena’s conjecture ([Mal], [Wi]), M d+1 is the anti de Sitter space
D.: Period functions and the Selberg zeta function for the modular group. The mathematical beauty of physics (Saclay
, 1996
"... The Selberg trace formula on a Riemann surface X connects the discrete spectrum of the Laplacian with the length spectrum of the surface, that is, the set of lengths of the closed geodesics of on X. The connection is most strikingly expressed in terms of the Selberg zeta function, which is a meromor ..."
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Cited by 30 (1 self)
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The Selberg trace formula on a Riemann surface X connects the discrete spectrum of the Laplacian with the length spectrum of the surface, that is, the set of lengths of the closed geodesics of on X. The connection is most strikingly expressed in terms of the Selberg zeta function, which is a meromorphic function of a complex variable s that is defined for ℜ(s)> 1 in terms of the length spectrum and that has zeros at all s ∈ C for which s(1 − s) is an eigenvalue of the Laplacian in L2 (X). We will be interested in the case when X is the quotient of the upper halfplane H by either the modular group Γ1 = SL(2, Z) or the extended modular group Γ = GL(2, Z), where γ = ( a b c d ∈ Γ acts on H by z ↦ → (az + b)/(cz + d) if det(γ) = +1 and z ↦ → (a¯z + b)/(c¯z + d) if det(γ) = −1. In this case the length spectrum of X is given in terms of class numbers and units of orders in real quadratic fields, while the eigenfunctions of the Laplace operator are the nonholomorphic modular functions usually called Maass wave forms. (Good expositions of this subject can be found in [6] and [7]). A striking fact, discovered by D. Mayer [4, 5] and for which a simplified proof will be given in the first part of this paper, is that the Selberg zeta function ZΓ(s) of H/Γ can be represented as the (Fredholm) determinant of the action of a certain element of the quotient field of the group ring Z[Γ] on an appropriate Banach space. Specifically, let V be the space of functions holomorphic in D = {z ∈ C  z − 1  < 3 2} and continuous in D. The ( a b) () −2s az+b semigroup {γ ∈ Γ  γ(D) ⊆ D} acts on the right by πs c d f(z) = (cz + d) f cz+d. In particular, for all n ≥ 1 the element ( 0 1), which can be written in terms of the generators σ = ( 1 0 1 1 expression and ρ =
Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators
 ALGORITHMICA
, 1998
"... We provide here a complete averagecase analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N. We analyze the three main parameters of the binary continued fraction expansion, namely, the height, the number of steps of ..."
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Cited by 24 (3 self)
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We provide here a complete averagecase analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N. We analyze the three main parameters of the binary continued fraction expansion, namely, the height, the number of steps of the binary Euclidean algorithm, and finally the sum of the exponents of powers of 2 contained in the numerators of the binary continued fraction. The average values of these parameters are shown to be asymptotic to Ai log N, and the three constants Ai are related to the invariant measure of the Perron–Frobenius operator linked to this dynamical system. The binary Euclidean algorithm was previously studied in 1976 by Brent who provided a partial analysis of the number of steps, based on a heuristic model and some unproven conjecture. Our methods are quite different, not relying on unproven assumptions, and more general, since they allow us to study all the parameters of the binary continued fraction expansion.
Transfer operator for Γ0(n) and the Hecke operators for the period functions
 of PSL(2, Z). Math. Proc. of the Cambridge Philosophical Society
"... In this article we report on a surprising relation between the transfer operators for the congruence subgroups Γ0(n) and the Hecke operators on the space of period functions for the modular group PSL(2, Z). For this we study special eigenfunctions of the transfer operators with eigenvalues ∓1, which ..."
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Cited by 21 (5 self)
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In this article we report on a surprising relation between the transfer operators for the congruence subgroups Γ0(n) and the Hecke operators on the space of period functions for the modular group PSL(2, Z). For this we study special eigenfunctions of the transfer operators with eigenvalues ∓1, which are also solutions of the Lewis equations for the groups Γ0(n) and which are determined by eigenfunctions of the transfer operator for the modular group PSL(2, Z). In the language of the AtkinLehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of Lewis equation. It turns out that the sum of the components of these old solutions for the group Γ0(n) determine for any n a solution of the Lewis equation for the modular group and hence also an eigenfunction of the transfer operator for this group. Our construction gives in this way linear operators in the space of period functions for the group PSL(2, Z). Indeed these operators are just the Hecke operators for the period functions of the modular group derived previously by Zagier and Mühlenbruch using the EichlerManinShimura correspondence between period polynomials and modular forms for the modular group. 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.
The Spectrum of Weakly Coupled Map Lattices
, 1997
"... We consider weakly coupled analytic expanding circle maps on the lattice Z d (for d ≥ 1), with small coupling strength ɛ and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (PerronFrobenius) transfer operators. We give a Fréchet space on ..."
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Cited by 15 (1 self)
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We consider weakly coupled analytic expanding circle maps on the lattice Z d (for d ≥ 1), with small coupling strength ɛ and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (PerronFrobenius) transfer operators. We give a Fréchet space on which the operator associated to the full system has a simple eigenvalue at 1 (corresponding to the SRB measure µɛ previously obtained by Bricmont–Kupiainen [BK1]) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For d = 1 we also construct Banach spaces of densities with respect to µɛ on which perturbation theory, applied to the difference of fixed high iterates of the normalised coupled and uncoupled transfer operators, yields localisation of the full spectrum of the coupled operator (i.e., the first spectral gap and beyond). As a sideeffect, we show that the whole spectra of the truncated coupled transfer operators (on bounded analytic functions) are O(ɛ)close to the truncated uncoupled spectra, uniformly in the spatial size. Our method uses polymer expansions and also gives the exponential decay of timecorrelations for a larger class of observables than those considered in [BK1].
On Khintchine exponents and Lyapunov exponents of continued fractions, Ergod
 Th. Dynam. Sys
"... Abstract. Assume that x ∈ [0,1) admits its continued fraction expansion x = [a1(x), a2(x), · · ·]. The Khintchine exponent γ(x) of x is defined by 1 Pn γ(x): = lim n→ ∞ n j=1 log aj(x) when the limit exists. Khintchine spectrum dim Eξ is fully studied, where Eξ: = {x ∈ [0, 1) : γ(x) = ξ} (ξ ≥ 0) ..."
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Cited by 15 (8 self)
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Abstract. Assume that x ∈ [0,1) admits its continued fraction expansion x = [a1(x), a2(x), · · ·]. The Khintchine exponent γ(x) of x is defined by 1 Pn γ(x): = lim n→ ∞ n j=1 log aj(x) when the limit exists. Khintchine spectrum dim Eξ is fully studied, where Eξ: = {x ∈ [0, 1) : γ(x) = ξ} (ξ ≥ 0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as function of ξ ∈ [0,+∞), is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by γϕ 1 Pn (x): = lim n→ ∞ ϕ(n) j=1 log aj(x) are also studied, where ϕ(n) tends to the infinity faster than n does. Under some regular conditions on ϕ, it is proved that the fast Khintchine spectrum dim({x ∈ [0,1] : γϕ (x) = ξ}) is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum.
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
Calculating Hausdorff Dimension Of Julia Sets And Kleinian Limit Sets
 Amer. J. Math
"... We present a new algorithm for efficiently computing the Hausdorff dimension of sets X invariant under conformal expanding dynamical systems. By locating the periodic points of period up to N , we construct approximations s N which converge to dim(X) superexponentially fast in N . This method can b ..."
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Cited by 15 (2 self)
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We present a new algorithm for efficiently computing the Hausdorff dimension of sets X invariant under conformal expanding dynamical systems. By locating the periodic points of period up to N , we construct approximations s N which converge to dim(X) superexponentially fast in N . This method can be used to give rigorous estimates for important examples, including hyperbolic Julia sets and limit sets of Schottky and quasifuchsian groups.