Results 1  10
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21
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 44 (15 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 29 (7 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 24 (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 =
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 15 (4 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
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 14 (9 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
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 14 (5 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 12 (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].
Intermittency and regularized Fredholm determinants
 Invent. Math
, 1999
"... We consider realanalytic maps of the interval I = [0,1] which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the spectrum of the associated PerronFrobenius operator M has a decomposition sp(M) = σc ∪ σp where σc = [0,1] is the continuous s ..."
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Cited by 10 (1 self)
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We consider realanalytic maps of the interval I = [0,1] which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the spectrum of the associated PerronFrobenius operator M has a decomposition sp(M) = σc ∪ σp where σc = [0,1] is the continuous spectrum of M and σp is the pure point spectrum with no points of accumulation outside 0 and 1. We construct a regularized Fredholm determinant d(λ) which has a holomorphic extension to λ ∈ C − σc and can be analytically continued from each side of σc to an open neighborhood of σc − {0,1} (on different Riemann sheets). In C − σc the zeroset of d(λ) is in onetoone correspondence with the point spectrum of M. Through the conformal transformation λ(z) = 1 4z (1 + z)2 the function d ◦ λ(z) extends to a holomorphic function in a domain which contains the unit disc. Shorttitle: Intermittency and Regularized Fredholm Determinants. 1 Assumptions and statement of results.
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 10 (1 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.
Orthonormal Expansions of Invariant Densities for Expanding Maps
 Adv. Math
, 2002
"... We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [U] of phase space discretisation. ..."
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Cited by 6 (4 self)
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We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [U] of phase space discretisation.