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17
Smooth Anosov flows: correlation spectra and stability
 J. Modern Dynamics
, 2007
"... Abstract. By introducing appropriate Banach spaces one can study the spectral properties of the generator of the semigroup defined by an Anosov flow. Consequently, it is possible to easily obtain sharp results on the Ruelle resonances and the differentiability of the SRB measure. 1. introduction I ..."
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Cited by 31 (7 self)
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Abstract. By introducing appropriate Banach spaces one can study the spectral properties of the generator of the semigroup defined by an Anosov flow. Consequently, it is possible to easily obtain sharp results on the Ruelle resonances and the differentiability of the SRB measure. 1. introduction In the last years there has been a growing interest in the dependence of the SRB measures on the parameters of the system. In particular, G.Gallavotti [11] has argued the relevance of such an issue for nonequilibrium statistical mechanics. On a physical basis (linear response theory) one expects that the average behaviour of an observable changes smoothly with parameters. Yet the related rigorous results are very limited and the existence of very irregular dependence from parameters (think, for example, to the quadratic family) shows that, in general, smooth dependence must be properly interpreted to have any chance to hold. The only cases in which some simple rigorous results are available are smooth uniformly hyperbolic systems and some partially hyperbolic systems. In particular,
Almost sure invariance principle for dynamical systems by spectral methods
 Ann. Probab
"... Abstract. We prove the almost sure invariance principle for stationary R d – valued processes (with dimensionindependent very precise error terms), solely under a strong assumption on the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical sy ..."
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Cited by 30 (0 self)
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Abstract. We prove the almost sure invariance principle for stationary R d – valued processes (with dimensionindependent very precise error terms), solely under a strong assumption on the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains, using strong or weak spectral perturbation arguments. The almost sure invariance principle is a very strong reinforcement of the central limit theorem: it ensures that the trajectories of a process can be matched with the trajectories of a brownian motion in such a way that, almost surely, the error between the trajectories is negligible compared to the size of the trajectory (the result can be more or less precise, depending on the specific error term one can obtain). This kind of results has a lot of consequences, see e.g. [MN09] and references therein. Such results are well known for onedimensional processes, either independent or weakly dependent (see among many others [DP84, HK82]), and for independent higher dimensional processes [Ein89, Zaĭ98]. However, for weakly dependent higher
Decay of correlations in suspension semiflows of anglemultiplying maps
, 2005
"... Abstract. We consider suspension semiflows of anglemultiplying maps on the circle. Under a C r generic condition on the ceiling function, we show that there exists an anisotropic Sobolev space[3] contained in the L 2 space such that the PerronFrobenius operator for the timetmap acts on it and t ..."
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Cited by 23 (2 self)
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Abstract. We consider suspension semiflows of anglemultiplying maps on the circle. Under a C r generic condition on the ceiling function, we show that there exists an anisotropic Sobolev space[3] contained in the L 2 space such that the PerronFrobenius operator for the timetmap acts on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description on decay of correlations and extends the result of M. Pollicott[17]. 1.
A simple framework to justify linear response theory
 NONLINEARITY
, 2009
"... The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced d ..."
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Cited by 20 (7 self)
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The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced dissipative stochastic dynamical systems is developed. The main results are formulated in an abstract setting and apply to suitable systems, in finite and infinite dimensions, that are of interest in climate change science and other applications.
Exponential decay of correlations for piecewise cone hyperbolic contact flows
 Comm. Math. Phys
, 2012
"... Abstract. We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuoustime dynamics with singularities on a manifold. Our proof combines ..."
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Cited by 17 (4 self)
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Abstract. We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuoustime dynamics with singularities on a manifold. Our proof combines the second author’s version [30] of Dolgopyat’s estimates for contact flows and the first author’s work with Gouëzel [6] on piecewise hyperbolic discretetime dynamics.
GOOD BANACH SPACES FOR PIECEWISE HYPERBOLIC MAPS VIA INTERPOLATION
, 2007
"... We introduce a weak transversality condition for piecewise C 1+α and piecewise hyperbolic maps which admit a C 1+α stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of TriebelLizorkin type. ..."
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Cited by 15 (2 self)
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We introduce a weak transversality condition for piecewise C 1+α and piecewise hyperbolic maps which admit a C 1+α stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of TriebelLizorkin type. In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) and applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results.
Multidimensional expanding maps with singularities: a pedestrian approach
 Ergod. Th. Dynam. Sys
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Entropic Fluctuations in Statistical Mechanics I. Classical Dynamical Systems
, 2010
"... Within the abstract framework of dynamical system theory we describe a general approach to the Transient (or EvansSearles) and Steady State (or GallavottiCohen) Fluctuation Theorems of nonequilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathemati ..."
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Cited by 5 (3 self)
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Within the abstract framework of dynamical system theory we describe a general approach to the Transient (or EvansSearles) and Steady State (or GallavottiCohen) Fluctuation Theorems of nonequilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. Besides its conceptual simplicity, another advantage of our approach is its natural extension to quantum statistical mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.
Alternative proofs of linear response for piecewise expanding unimodal maps
, 2008
"... We give two new proofs that the SRB measure t ↦ → µt of a C² path ft of unimodal piecewise expanding C³ maps is differentiable at 0 if ft is tangent to the topological class of f0. The arguments are more conceptual than the one in [4], but require proving Hölder continuity of the infinitesimal con ..."
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Cited by 3 (2 self)
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We give two new proofs that the SRB measure t ↦ → µt of a C² path ft of unimodal piecewise expanding C³ maps is differentiable at 0 if ft is tangent to the topological class of f0. The arguments are more conceptual than the one in [4], but require proving Hölder continuity of the infinitesimal conjugacy α (a new result, of independent interest) and using spaces of bounded pvariation. The first new proof gives differentiability of higher order of R ψ dµt if ft is smooth enough and stays in the topological class of f0 and if ψ smooth enough (a new result). In addition, this proof does not require any information on the decomposition of the SRB measure into regular and singular terms, making it potentially amenable to extensions to higher dimensions. The second new proof allows us to recover the linear response formula (i.e., the formula for the derivative at 0) obtained in [4], and gives additional information on this formula.
Linear Response, Or Else
"... Consider a smooth oneparameter family t 7 → ft of dynamical systems ft, with t  < . Assume that for all t (or for many t close to t = 0) the map ft admits a unique SRB invariant probability measure µt. We say that linear response holds if t 7 → µt is differentiable at t = 0 (possibly in the s ..."
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Cited by 1 (1 self)
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Consider a smooth oneparameter family t 7 → ft of dynamical systems ft, with t  < . Assume that for all t (or for many t close to t = 0) the map ft admits a unique SRB invariant probability measure µt. We say that linear response holds if t 7 → µt is differentiable at t = 0 (possibly in the sense of Whitney), and if its derivative can be expressed as a function of f0, µ0, and ∂tftt=0. The goal of this note is to present to a general mathematical audience recent results and open problems in the theory of linear response for chaotic dynamical systems, possibly with bifurcations.