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69
Determining Lyapunov Exponents from a Time Series
 Physica
, 1985
"... We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of n ..."
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Cited by 495 (1 self)
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We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the longterm growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the BelousovZhabotinskii reaction and CouetteTaylor flow. Contents 1.
Hausdorff Dimension of Measures via Poincaré Recurrence
, 2001
"... . We study the quantitative behavior of Poincare recurrence. ..."
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Cited by 38 (11 self)
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. We study the quantitative behavior of Poincare recurrence.
Harmonic measures versus quasiconformal measures for hyperbolic groups
, 2008
"... We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on ..."
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Cited by 30 (2 self)
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We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.
Random Series In Powers Of Algebraic Integers: Hausdorff Dimension Of The Limit Distribution
, 1995
"... We study the distributions F`;p of the random sums P 1 1 " n` n , where " 1 ; " 2 ; : : : are i.i.d. Bernoullip and ` is the inverse of a Pisot number (an algebraic integer fi whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = :5, F`;p ..."
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Cited by 27 (0 self)
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We study the distributions F`;p of the random sums P 1 1 " n` n , where " 1 ; " 2 ; : : : are i.i.d. Bernoullip and ` is the inverse of a Pisot number (an algebraic integer fi whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = :5, F`;p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain fi of small degree, simulation gives the Hausdorff dimension to several decimal places.
Lyapunov exponents for products of matrices and multifractal analysis, Part I: Positive matrices
 Israel J. Math
"... Abstract. We continue the study in [11, 14] on the upper Lyapunov exponents for products of matrices. Here we consider general matrices. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. Anyway we focus our interest on a special case w ..."
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Cited by 24 (9 self)
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Abstract. We continue the study in [11, 14] on the upper Lyapunov exponents for products of matrices. Here we consider general matrices. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. Anyway we focus our interest on a special case where the matrix function M(x) takes finite values M1,..., Mm. In this case we prove the variational formula under an additional irreducibility condition. This extends a previous result of the author and Lau [14]. As an application, we prove a new multifractal formalism for a certain class of selfsimilar measures on R with overlaps. More precisely, let µ be the selfsimilar measure on R generated by a family of contractive similitudes {Sj = ρx + bj}`j=1 which satisfies the finite type condition. Then we can construct a family (finite or countably infinite) of closed intervals {Ij}j∈Λ with disjoint interiors, such that µ is supported on j∈Λ Ij and the restricted measure µIj of µ on each interval Ij satisfies the complete multifractal formalism. Moreover the dimension spectrum dimH EµIj (α) is independent of j. 1.
A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups, Ergodic theory and dynamical systems 24
, 2004
"... ABSTRACT. We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these gro ..."
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Cited by 22 (12 self)
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ABSTRACT. We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these groups our revised formalisms give access to a description of the spectrum of ‘homological growth rates ’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ’strong phase transitions’. 1.
On the limit Rademacher functions and Bernoulli convolutions, in: Dynamical Systems
 Proceedings of the International Conference in Honor of Professor Liao Shantao, World Scientific
, 1999
"... convolutions associated with Pisot numbers ..."
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Dimension and product structure of hyperbolic measures
 ANNALS OF MATHEMATICS, 149 (1999), 755–783
, 1999
"... We prove that every hyperbolic measure invariant under a C 1+α diffeomorphism of a smooth Riemannian manifold possesses asymptotically “almost” local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. Th ..."
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Cited by 17 (0 self)
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We prove that every hyperbolic measure invariant under a C 1+α diffeomorphism of a smooth Riemannian manifold possesses asymptotically “almost” local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. This has not been known even for measures supported on locally maximal hyperbolic sets. Using this property of hyperbolic measures we prove the longstanding EckmannRuelle conjecture in dimension theory of smooth dynamical systems: the pointwise dimension of every hyperbolic measure invariant under a C 1+α diffeomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide. This provides the rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures.
Hausdorff Dimension Of The Harmonic Measure On Trees
 Systems
, 1997
"... . For a large class of Markov operators on trees we prove the formula HD = h=l connecting the Hausdorff dimension of the harmonic measure on the tree boundary, the rate of escape l and the asymptotic entropy h. Applications of this formula include random walks on free groups, conditional random w ..."
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Cited by 15 (5 self)
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. For a large class of Markov operators on trees we prove the formula HD = h=l connecting the Hausdorff dimension of the harmonic measure on the tree boundary, the rate of escape l and the asymptotic entropy h. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations. 0. Introduction The Hausdorff dimension HD¯ of a measure ¯ on a metric space (X; d) is defined as the minimal Hausdorff dimension of sets of full measure ¯ and shows the "degree of singularity" (or, of "fractalness" in the newspeak) of this measure. Even if the support of the measure ¯ is the whole space, HD¯ does not have to coincide with HDX. The Hausdorff dimension HD¯ characterizes the polynomial rate of decreasing of the measures ¯ of balls of the metric d around typical (with respect to ¯) points of X, in particular, if ball measures decrease regularly, i.e., the limit lim log ¯B(x; r)= log r = f...
Falconer's Formula for the Hausdorff Dimension of a SelfAffine Set in R²
 IN R
, 1993
"... Simple sufficient conditions are given for the validity of a formula of Falconer [3] describing the Hausdorff dimension of a selfaffine set. These conditions are natural (and easily checked) geometric restrictions on the actions of the affine mappings determining the selfaffine set. It is also sho ..."
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Cited by 15 (1 self)
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Simple sufficient conditions are given for the validity of a formula of Falconer [3] describing the Hausdorff dimension of a selfaffine set. These conditions are natural (and easily checked) geometric restrictions on the actions of the affine mappings determining the selfaffine set. It is also shown that under these hypotheses the selfaffine set supports an invariant Gibbs measure whose Hausdorff dimension equals that of the set.