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91
Period three implies chaos
 Amer. Math. Monthly
, 1975
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 160 (0 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
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 124 (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...
Adaptive Nonlinear Approximations
, 1994
"... The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing the dictionary waveforms which best matc ..."
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Cited by 52 (1 self)
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The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing the dictionary waveforms which best match the function's structures. Matching pursuits provide a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and using a stochastic differential equation model. These invariant measures define a noise with respect to a given dictionary. ...
Bulls, bears and market sheep
 Journal of Economic Behavior & Organization
, 1990
"... A deterministic excess demand model of stock market behavior is presented that generates stochastically fluctuating prices and randomly switching bear and bull markets. 1. ..."
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Cited by 47 (0 self)
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A deterministic excess demand model of stock market behavior is presented that generates stochastically fluctuating prices and randomly switching bear and bull markets. 1.
Set Oriented Numerical Methods for Dynamical Systems
, 2000
"... Contents 1 Introduction 1 2 The Computation of Invariant Sets 2 2.1 Brief Review on Invariant Sets . . . . . . . . . . . . . . . . . . 2 2.2 The Computation of Relative Global Attractors . . . . . . . . 3 2.3 Convergence Behavior and Error Estimate . . . . . . . . . . . 6 2.4 Numerical Examples . ..."
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Cited by 36 (12 self)
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Contents 1 Introduction 1 2 The Computation of Invariant Sets 2 2.1 Brief Review on Invariant Sets . . . . . . . . . . . . . . . . . . 2 2.2 The Computation of Relative Global Attractors . . . . . . . . 3 2.3 Convergence Behavior and Error Estimate . . . . . . . . . . . 6 2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 The Computation of Chain Recurrent Sets . . . . . . . . . . . 9 2.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The Computation of Invariant Manifolds 12 3.1 Description of the Method . . . . . . . . . . . . . . . . . . . . 13 3.2 Convergence Behavior and Error Estimate . . . . . . . . . . . 14 3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 15 4 The Computation of SRBMeasures 18 4.1 Brief Review on SRBMeasures and Small Random Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Spectral Approximation for the Per
Stability of the Spectrum for Transfer Operators
 ANN. SCUOLA NORM. SUP. PISA CL SCI
, 1998
"... We prove stability of the isolated eigenvalues of transfer operators satisfying a LasotaYorke type inequality under a broad class of random and nonrandom perturbations including Ulamtype discretizations. The results are formulated in an abstract framework. ..."
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Cited by 30 (5 self)
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We prove stability of the isolated eigenvalues of transfer operators satisfying a LasotaYorke type inequality under a broad class of random and nonrandom perturbations including Ulamtype discretizations. The results are formulated in an abstract framework.
Statistical stability for robust classes of maps with nonuniform expansion, Ergd
 Th. & Dynam. Sys
"... We consider open sets of maps in a manifold M exhibiting nonuniform expanding behaviour in some domain S ⊂ M. Assuming that there is a forward invariant region containing S where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuousl ..."
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Cited by 27 (11 self)
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We consider open sets of maps in a manifold M exhibiting nonuniform expanding behaviour in some domain S ⊂ M. Assuming that there is a forward invariant region containing S where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuously in the L 1norm with the map. As a main application we show that the open class of maps introduced in [V] fits to this situation, thus proving that the SRB measures constructed in [A] vary continuously with the map. 1
Strong Stochastic Stability and Rate of Mixing for Unimodal Maps
 Ann. Sci. ' Ecole Norm. Sup
, 1994
"... . We consider small random perturbations of a large class of nonuniformly hyperbolic unimodal maps and prove stochastic stability in the strong sense (L 1 convergence of invariant densities) and uniform bounds for the exponential rate of decay of correlations. Our method is based on an analysis o ..."
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Cited by 26 (5 self)
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. We consider small random perturbations of a large class of nonuniformly hyperbolic unimodal maps and prove stochastic stability in the strong sense (L 1 convergence of invariant densities) and uniform bounds for the exponential rate of decay of correlations. Our method is based on an analysis of the spectrum of a modified PerronFrobenius operator for a tower extension of the Markov chain. 1. Introduction Let I ae R be a compact interval and f : I ! I be a smooth unimodal map with f(I) ae int (I). The prototype we have in mind are the quadratic maps f(x) = \Gammax 2 + a but our arguments and conclusions hold in the general context of maps with negative Schwarzian derivative and nondegenerate critical point. Let c 2 I be the critical point of f and c k = f k (c) for k 0. Throughout this paper we assume that (A1) jf k (c) \Gamma cj e \Gammaffk for all k H 0 , (A2) j(f k ) 0 (c 1 )j k c for all k H 0 , (A3) f is topologically mixing on the interval bounded by c ...
Decay Of Correlations For Piecewise Expanding Maps
 JOURNAL OF STATISTICAL PHYSICS
, 1995
"... This paper investigates the decay of correlations in a large class of nonMarkov onedimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained. ..."
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Cited by 19 (4 self)
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This paper investigates the decay of correlations in a large class of nonMarkov onedimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained.
Absolutely Continuous Invariant Measures for Multidimensional Expanding Maps
, 1998
"... We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps w ..."
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Cited by 17 (2 self)
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We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps with singularities. We allow maps that are discontinuous on some extremely wild sets, the shape of the discontinuities being completely ignored with our approach. KeyWords: Expanding maps with singularities, Invariant measure, Decay of correlations, PerronFrobenius operator. October 1997 Second version December 1998 CPT97/P.3552 anonymous ftp : ftp.cpt.univmrs.fr web : www.cpt.univmrs.fr Unite Propre de Recherche 7061 1 PhyMat, Mathematics Department, University of Toulon. 83957 La Garde, France. Invariant Measure for Multidimensional Expanding Maps 1 Contents 1 Introduction 1 2 Piecewise expanding maps 4 3 QuasiHolder space 9 4 A LasotaYorke type Inequality 12 5 Spectral res...