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35
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 ...
The thermodynamic formalism approach to Selberg’s zeta function for PSL(2,Z
 Bull. Amer. Math. Soc
, 1991
"... Besides the classical approach to Selberg's zeta function for cofinite Fuchsian groups [S] through the trace formula [V] there has been developed recently another one based on the thermodynamic formalism [R2] applied to the dynamical zeta function of Smale ..."
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Cited by 22 (4 self)
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Besides the classical approach to Selberg's zeta function for cofinite Fuchsian groups [S] through the trace formula [V] there has been developed recently another one based on the thermodynamic formalism [R2] applied to the dynamical zeta function of Smale
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 17 (4 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
Periodic Orbits as the Skeleton of Classical and Quantum Chaos
 PROCEEDINGS OF THE LOS ALAMOS CENTER FOR NONLINEAR SCIENCE NONLINEAR SCIENCE  NEXT DECADE
, 1990
"... A description of a lowdimensional deterministic chaotic system in terms of unstable periodic orbits (cycles) is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos, comparable to the familiar perturbation expansions for nearly integrable syste ..."
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Cited by 16 (3 self)
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A description of a lowdimensional deterministic chaotic system in terms of unstable periodic orbits (cycles) is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos, comparable to the familiar perturbation expansions for nearly integrable systems. The infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a Selberg product form, dominated by the short cycles, and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits. Methods for implementing this computation for finite subshift dynamics are introduced.
Average BitComplexity of Euclidean Algorithms
 Proceedings ICALP’00, Lecture Notes Comp. Science 1853, 373–387
, 2000
"... We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set ..."
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Cited by 15 (6 self)
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We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where 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 algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of an entire class of gcdlike algorithms. Keywords: Averagecase Analysis of algorithms, BitComplexity, Euclidean Algorithms, Dynamical Systems, Ruelle operators, Generating Functions, Dirichlet Series, Tauberian Theorems. 1 Introduction Motivations. Euclid's algorithm was analysed first in the worst case in 1733 by de Lagny, then in the averagecase around 1969 independently by Heilbronn [12] and Dixon [6], and finally in distribution by Hensley [13] who proved in 1994 that the Eu...
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.
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.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
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.
Infinite Kneading Matrices And Weighted Zeta Functions Of Interval Maps
 J. Functional Analysis
, 1994
"... . We consider a piecewise continuous, piecewise monotone interval map and a weight of bounded variation, constant on homtervals and continuous at periodic points of the map. With these data we associate a sequence of weighted MilnorThurston kneading matrices, converging to a countable matrix with c ..."
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Cited by 9 (2 self)
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. We consider a piecewise continuous, piecewise monotone interval map and a weight of bounded variation, constant on homtervals and continuous at periodic points of the map. With these data we associate a sequence of weighted MilnorThurston kneading matrices, converging to a countable matrix with coefficients analytic functions. We show that the determinants of these matrices converge to the inverse of the correspondingly weighted zeta function for the map. As a corollary, we obtain convergence of the discrete spectrum of the PerronFrobenius operators of piecewise linear approximations of Markovian, piecewise expanding and piecewise C 1+BV interval maps. Introduction Let f be a transformation of a compact interval, say I = [0; 1]. Assume that f is piecewise continuous, and piecewise strictly monotone, with a finite number N of pieces defined by turning points 0 = a 0 ! a 1 ! a 2 ! : : : ! aN \Gamma1 ! aN = 1. Let g : I ! C be a weight (a natural choice is g = 1=jf 0 j, if f is ...