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18
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 40 (7 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...
On The Spectra Of Randomly Perturbed Expanding Maps
 Comm. Math. Phys
, 1993
"... . We consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing. We do this by proving the robustness of the spectra of their PerronFrobenius operators. Introduction Let f : M !M be a dynamical system pres ..."
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Cited by 38 (11 self)
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. We consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing. We do this by proving the robustness of the spectra of their PerronFrobenius operators. Introduction Let f : M !M be a dynamical system preserving some natural probability measure ¯ 0 with density ae 0 . This paper is motivated by the following question: does exponential mixing imply stochastic stability? Roughly speaking, exponential mixing of (f; ¯ 0 ) means that, for two observables ' and / on M , the correlation between ' ffi f n and / decays exponentially fast with n. Stochastic stability means that, if we add a small amount of random noise to f , obtaining at noise level ffl a Markov process with invariant density ae ffl , then ae ffl tends to ae 0 as ffl tends to zero. The following heuristic argument suggests an affirmative answer to this question. Consider the PerronFrobenius operator L associated with f acting on ...
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 operat ..."
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Cited by 28 (4 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 Sources in Information Theory: Fundamental intervals and Word Prefixes.
, 1998
"... A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some important problems about prefixes that intervene in algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewe ..."
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Cited by 28 (7 self)
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A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some important problems about prefixes that intervene in algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewed as a "generating" operator. Its dominant spectral objects are linked with important parameters of the source such as the entropy, and play a central role in all the results. 1 Introduction. In information theory contexts, data items are (infinite) words that are produced by a common mechanism, called a source. Realistic sources are often complex objects. We work here inside a quite general framework of sources related to dynamical systems theory which goes beyond the cases of memoryless and Markov sources. This model can describe nonmarkovian processes, where the dependency on past history is unbounded, and as such, they attain a high level of generality. A probabilistic dynamical source ...
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
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.
Computing The Dimension Of Dynamically Defined Sets I: E2 and Bounded . . .
"... We present a powerful approach to computing the Hausdorff dimension of certain conformally selfsimilar sets. We illustrate this method for the dimension dim H (E 2 ) of the set E 2 , consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking ..."
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Cited by 11 (2 self)
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We present a powerful approach to computing the Hausdorff dimension of certain conformally selfsimilar sets. We illustrate this method for the dimension dim H (E 2 ) of the set E 2 , consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking feature of this method is that the successive approximations converge to dim(E 2 ) at a superexponential rate.
Continued Fractions, Comparison Algorithms, and Fine Structure Constants
, 2000
"... There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems ..."
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Cited by 10 (2 self)
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There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems (symbolic dynamics), number theory (continued fractions), special functions (multiple zeta values), functional analysis (transfer operators), numerical analysis (series acceleration), and complex analysis (the Riemann hypothesis). These domains all eventually contribute to a detailed characterization of the complexity of comparison and sorting algorithms, either on average or in probability.
SYMBOLIC DYNAMICS FOR THE MODULAR SURFACE AND BEYOND
, 2007
"... In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording ..."
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Cited by 7 (0 self)
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In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.
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 6 (5 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.