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189
Statistical Properties of Dynamical Systems with Some Hyperbolicity
, 1996
"... MODEL AND ITS MIXING PROPERTIES 1. Setting and Assertions Let f : M \Psi be a C 1+" diffeomorphism of a finite dimensional Riemannian manifold M . In applications we will allow f to have discontinuities or singularities, but these "bad" parts will not appear in the picture we are about to describ ..."
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Cited by 125 (7 self)
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MODEL AND ITS MIXING PROPERTIES 1. Setting and Assertions Let f : M \Psi be a C 1+" diffeomorphism of a finite dimensional Riemannian manifold M . In applications we will allow f to have discontinuities or singularities, but these "bad" parts will not appear in the picture we are about to describe. Thus as far as Part I is concerned we may assume that f and f \Gamma1 are defined on all of M . Let d(\Delta; \Delta) denote the distance between points. Riemannian measure on M will be denoted by ; and if W ae M is a submanifold, then W denotes the measure on W induced by the restriction of the Riemannian structure to W . The basic object of interest here consists of a set ae M with a "hyperbolic product structure" and a return map f R from to itself. Precise definitions are given in 1.1 and 1.2; the 4 required properties are listed in (P1)(P5); and the main results of Part I are stated in 1.4. 1.1. A "horseshoe" with infinitely many branches and variable return times. We be...
Recurrence Times And Rates Of Mixing
, 1997
"... The setting of this paper consists of a map making "nice" returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem are given in terms of the tail of the return time function. The abstract setting considered arises natu ..."
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Cited by 101 (5 self)
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The setting of this paper consists of a map making "nice" returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem are given in terms of the tail of the return time function. The abstract setting considered arises naturally in differentiable dynamical systems with some expanding or hyperbolic properties. This paper is part of an attempt to understand the speed of mixing and related statistical properties for chaotic dynamical systems. More precisely, we are interested in systems that are expanding or hyperbolic on large parts (though not necessarily all) of their phase spaces. A natural approach to this problem is to pick a suitable reference set, and to regard a part of the system as having "renewed" itself when it makes a "full" return to this set. We obtain in this way a representation of the dynamical system in question, described in terms of a reference set and return times. We propose to study thi...
Dimensions and measures in infinite iterated function systems
 PROC. LONDON MATH. SOC
, 1996
"... The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to ..."
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Cited by 83 (21 self)
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The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to be unique. The existence of a unique invariant probability equivalent to the conformal measure is derived. Our methods employ the concepts of the PerronFrobenius operator, symbolic dynamics on an infinite dimensional shift space, and the properties of the above mentioned ergodic invariant measure. A formula for the Hausdorff dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by finite systems are shown to appear in the infinite case. In particular a variety of conditions are provided for Hausdorff and packing measures to be positive or finite, and a number of examples are described showing the appearance of various possible combinations for these quantities. One example given special attention is the limit set associated to the complex continued fraction expansion  in particular lower and upper estimates for its Hausdor dimension are given. A large natural class of systems whose limit sets are "dimensionless in the restricted sense" is described.
Dynamical Sources in Information Theory: A General Analysis of Trie Structures
 ALGORITHMICA
, 1999
"... Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of arraytries, list tries, and bsttries ("ternary search tries"). The size and the sear ..."
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Cited by 50 (7 self)
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Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of arraytries, list tries, and bsttries ("ternary search tries"). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of height of tries is given. The unifying data model used is that of dynamical sources and it encompasses classical models like those of memoryless sources with independent symbols, of finite Markovchains, and of nonuniform densities. The probabilistic behaviour of the main parameters, namely size, path length, or height, appears to be determined by two intrinsic characteristics of the source: the entropy and the probability of letter coincidence. These characteristics are themselves related in a natural way to spectral properties of specific transfer operators of the Ruelle type.
High Temperature Expansions and Dynamical Systems
 Comm. Math. Phys
, 1996
"... We develop a resummed hightemperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the Pe ..."
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Cited by 39 (2 self)
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We develop a resummed hightemperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the PerronFrobenius operator of weakly coupled map lattices. 1 Introduction. The theory of Gibbs states was originally developed for the mathematical analysis of equilibrium statistical mechanics. An interesting application of the theory was found by Sinai, Ruelle and Bowen in the 70's [41, 42, 38, 1] who applied it to the ergodic theory of uniformly hyperbolic dynamical systems. While this so called thermodynamic formalism has been very successful in ergodic theory, the Gibbs states that describe the statistics of such dynamical systems are quite simple from the point of view of statistical mechanics: they describe one dimensional spin systems with spins taking values in a finite set and intera...
Lowtemperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitelymany ground states
"... Starting from classical lattice systems in d 2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that the addition of a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase ..."
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Cited by 31 (13 self)
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Starting from classical lattice systems in d 2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that the addition of a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase diagrams. The quantum perturbations can involve bosons or fermions and Present address: Facultad de Matem'atica, Astronom'ia y F'isica, Universidad Nacional de C'ordoba, Ciudad Universitaria, 5000 C'ordoba, Argentina. Email: fernande@fis.uncor.edu can be of infinite range but decaying exponentially fast with the size of the bonds. For fermions, the interactions must be given by monomials of even degree in creation and annihilation operators. Our methods can be applied to some anyonic systems as well. Our analysis is based on an extension of PirogovSinai theory to contour expansions in d + 1 dimensions obtained by iteration of the Duhamel formula. Keywords: Phase diagrams; quantum latti...
Analysis of the binary Euclidean algorithm
 Directions and Recent Results in Algorithms and Complexity
, 1976
"... The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids multiplications and divisions, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. We describe the binary algorithm and consider its average case behaviour. In ..."
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Cited by 28 (2 self)
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The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids multiplications and divisions, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. We describe the binary algorithm and consider its average case behaviour. In particular, we correct some errors in the literature, discuss some recent results of Vallée, and describe a numerical computation which supports a conjecture of Vallée. 1
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 ...
A.: Almost Gibbsian versus weakly Gibbsian measures
 Stoch. Proc. Appl
, 1999
"... We consider two possible extensions of the standard de nition of Gibbs measures for lattice spin systems. When a random eld has conditional distributions which are almost surely continuous (almost Gibbsian eld), then there is a potential for that eld which is almost surely summable (weakly Gibbsian ..."
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Cited by 27 (11 self)
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We consider two possible extensions of the standard de nition of Gibbs measures for lattice spin systems. When a random eld has conditional distributions which are almost surely continuous (almost Gibbsian eld), then there is a potential for that eld which is almost surely summable (weakly Gibbsian eld). This generalizes the standard Kozlov theorems. The converse is not true in general as is illustrated by counterexamples.
On The Dimension Of Deterministic And Random CantorLike Sets
 Comm. Math. Phys
, 1994
"... this paper we unify and extend many of the known results on the Hausdorff and box dimension of deterministic and random Cantorlike sets in R determined by geometric constructions (see [PW] for the complete description of results and detailed proofs). Most authors have considered similarity proce ..."
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Cited by 24 (4 self)
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this paper we unify and extend many of the known results on the Hausdorff and box dimension of deterministic and random Cantorlike sets in R determined by geometric constructions (see [PW] for the complete description of results and detailed proofs). Most authors have considered similarity processes which impose a strong restriction on the geometry of the construction. Moreover, these constructions were modeled by either the full shift, or subshifts of finite type. In this paper we weaken these restrictions significantly and consider geometric constructions which need not be selfsimilar and have more complicated geometry. Our constructions are also modeled by arbitrary symbolic dynamical systems. Symbolic dynamics and the thermodynamic formalism thus become essential tools in our analysis. We also introduce two new fundamental classes of geometric constructions: asymptotic constructions and random constructions determined by an arbitrary ergodic stationary process