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124
SRB measures for partially hyperbolic systems whose central direction is mostly expanding
, 2000
"... We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms -- the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting -- under the assumption that the complementary subbundle is non-uniformly expanding. If the r ..."
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Cited by 197 (44 self)
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We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms -- the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting -- under the assumption that the complementary subbundle is non-uniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle). 1 Introduction The following approach has been most effective in studying the dynamics of complicated systems: one tries to describe the average time spent by typical orbits in different regions of the phase space. According to the ergodic theorem of Birkhoff, such times are well defined for almost all point, with respect to any invariant probability measure. However, the...
Escape rates and conditionally invariant measures
, 2005
"... We consider dynamical systems on domains that are not invariant under the dynamics – for example, a system with a hole in the phase space – and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, ..."
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Cited by 41 (6 self)
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We consider dynamical systems on domains that are not invariant under the dynamics – for example, a system with a hole in the phase space – and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, we are led quickly to conditionally invariant measures that are absolutely continuous with respect to Lebesgue (a.c.c.i.m.). Comparisons with SRB measures are inevitable, yet there are important differences. Via informal discussions and examples, this paper seeks to clarify the ideas involved. It includes also a brief review of known results and possible directions of further work in this developing subject.
Invariant Measures and Their Properties. A Functional Analytic Point of View
, 2002
"... In this series of lectures I try to illustrate systematically what I call the \functional analytic approach" to the study of the statistical properties of Dynamical Systems. The ideas are presented via a series of examples of increasing complexity, hoping to give in this way a feeling of the ..."
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Cited by 28 (2 self)
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In this series of lectures I try to illustrate systematically what I call the \functional analytic approach" to the study of the statistical properties of Dynamical Systems. The ideas are presented via a series of examples of increasing complexity, hoping to give in this way a feeling of the breadth of the method.
On the susceptibility function of piecewise expanding interval maps
- Comm. Math. Phys
"... n=0 n X(y)ρ0(y) ∂ ∂y ϕ(fn (y)) dy associated to the perturbation ft = f + tX of a piecewise expanding interval map f, and to an observable ϕ. The analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions on f, X, and ϕ which guarantee that Ψ(z) i ..."
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Cited by 27 (11 self)
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n=0 n X(y)ρ0(y) ∂ ∂y ϕ(fn (y)) dy associated to the perturbation ft = f + tX of a piecewise expanding interval map f, and to an observable ϕ. The analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions on f, X, and ϕ which guarantee that Ψ(z) is holomorphic in a disc of larger than one. Although R Ψ(1) is the formal derivative (at t = 0) of the average R(t) = ϕρt dx of ϕ with respect to the SRB measure of ft, we present examples of f, X, and ϕ satisfying our conditions so that R(t) is not Lipschitz at 0. that the set {x ∈ M | limn→ ∞ 1 n 1. Introduction and
Linear response formula for piecewise expanding unimodal maps
, 2007
"... The average R(t) = R ϕ dµt of a smooth function ϕ with respect to the SRB measure µt of a smooth one-parameter family ft of piecewise expanding interval maps is not always Lipschitz [4], [17]. We prove that if ft is tangent to the topological class of f, and if ∂tft|t=0 = X ◦ f, then R(t) is diff ..."
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Cited by 22 (7 self)
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The average R(t) = R ϕ dµt of a smooth function ϕ with respect to the SRB measure µt of a smooth one-parameter family ft of piecewise expanding interval maps is not always Lipschitz [4], [17]. We prove that if ft is tangent to the topological class of f, and if ∂tft|t=0 = X ◦ f, then R(t) is differentiable at zero, and R ′(0) coincides with the resummation proposed in [4] of the (a priori divergent) series P∞ R n=0 X(y)∂y(ϕ ◦ fn)(y) dµ0(y) given by Ruelle’s conjecture. In fact, we show that t ↦ → µt is differentiable within Radon measures. It is the first time that a linear response formula is obtained in a setting where structural stability does not hold. Violation of causality [25] reflects the fact that ft may be transversal to the topological class of f. that the set {x ∈ M | limn→ ∞ 1 n 1.
Fluctuation relations for diffusion process
- Commun. Math. Phys
"... The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-rever ..."
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Cited by 21 (6 self)
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The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations. 1
Strange attractors in periodically-kicked limit cycles and Hopf bifurcations
- Comm. Math. Phys
"... We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations. This paper is ab ..."
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Cited by 20 (12 self)
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We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations. This paper is about a mechanism for producing chaos. The scheme consists of periodic kicks interspersed with long periods of relaxation. We apply it to some very tame dynamical settings, such as limit cycles and stable equilibria undergoing Hopf bifurcations, and prove the appearance of chaotic behavior under reasonable conditions. The results in this paper, beginning with the statements in Section 1, are rigorous. The rest of this introduction is devoted to a nontechnical discussion of some of the ideas and issues surrounding this work. Main results In Theorem 1, we prove that when suitably kicked, all limit cycles can be turned into strange attractors with strong stochastic properties.
New Approximations and Tests of Linear Fluctuation-Response for Chaotic Nonlinear Forced-Dissipative Dynamical Systems
"... We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation-dissipation theorem. Unlike the earlier work in developing fluctuationdissipation theorem-type computational strategies ..."
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Cited by 19 (14 self)
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We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation-dissipation theorem. Unlike the earlier work in developing fluctuationdissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai-Ruelle-Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forceddissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation-dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.
EXTREME VALUE LAWS IN DYNAMICAL SYSTEMS FOR NON-SMOOTH OBSERVATIONS
, 2010
"... We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely conti ..."
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Cited by 18 (5 self)
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We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.
Stochastic Climate Dynamics: Random Attractors and Time-dependent Invariant Measures
, 2010
"... This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detai ..."
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Cited by 18 (8 self)
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This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detailed geometric structure of the random attractors associated with nonlinear, stochastically perturbed systems. A high-resolution numerical study of two highly idealized models of fundamental interest for climate dynamics allows one to obtain a good approximation of their global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to be random Sinai-Ruelle-Bowen (SRB) measures. The first of the two models is a stochastically forced version of the classical Lorenz model. The second one is a low-dimensional, nonlinear stochastic model of the El Niño–Southern Oscillation (ENSO).
