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41
RuellePerronFrobenius Spectrum For Anosov Maps
 Nonlinearity
, 2001
"... We extend a number of results from one dimensional dynamics based on spectral properties of the RuellePerronFrobenius transfer operator to Anosov di#eomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show ..."
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Cited by 33 (9 self)
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We extend a number of results from one dimensional dynamics based on spectral properties of the RuellePerronFrobenius transfer operator to Anosov di#eomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem. 1.
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 31 (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.
Central limit theorem and stable laws for intermittent maps
, 2002
"... In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to onedimensional maps with a neutral fixed point at 0 of the form x + x 1+α, for α ∈ (0,1). In particular, for α> 1/2, we show that the Bir ..."
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Cited by 26 (7 self)
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In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to onedimensional maps with a neutral fixed point at 0 of the form x + x 1+α, for α ∈ (0,1). In particular, for α> 1/2, we show that the Birkhoff sums of a Hölder observable f converge to a normal law or a stable law, depending on whether f(0) = 0 or f(0) ̸ = 0. The proof uses spectral techniques introduced by Sarig, and Wiener’s Lemma in noncommutative Banach algebras. 1 Introduction and statement of results
Almost sure invariance principle for nonuniformly hyperbolic systems
 COMM. MATH. PHYS
, 2005
"... We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizo ..."
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Cited by 21 (5 self)
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We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon. Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.
On contact Anosov flows
 Ann. of Math
"... Abstract. Exponential decay of correlations for C 4 Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature. 1. ..."
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Cited by 18 (0 self)
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Abstract. Exponential decay of correlations for C 4 Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature. 1.
Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as ..."
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Cited by 16 (8 self)
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We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinitedimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
Anisotropic Sobolev spaces and dynamical transfer operators: C1+α foliations, in preparation
"... Abstract. We consider a C ∞ Anosov diffeomorphism T with a C ∞ stable dynamical foliation. We show upper bounds on the essential spectral radius of its transfer operator acting on anisotropic Sobolev spaces. (Such bounds are related to the essential decorrelation rate for the SRB measure.) We compar ..."
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Cited by 16 (4 self)
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Abstract. We consider a C ∞ Anosov diffeomorphism T with a C ∞ stable dynamical foliation. We show upper bounds on the essential spectral radius of its transfer operator acting on anisotropic Sobolev spaces. (Such bounds are related to the essential decorrelation rate for the SRB measure.) We compare our results to the estimates of Kitaev on the domain of holomorphy of dynamical determinants for differentiable dynamics. 1.
Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms
, 2005
"... We study spectral properties of transfer operators for diffeomorphisms T: X → X on a Riemannian manifold X: Suppose that Ω is an isolated hyperbolic subset for T, with a compact isolating neighborhood V ⊂ X. We first introduce Banach spaces of distributions supported on V, which are anisotropic vers ..."
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Cited by 13 (5 self)
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We study spectral properties of transfer operators for diffeomorphisms T: X → X on a Riemannian manifold X: Suppose that Ω is an isolated hyperbolic subset for T, with a compact isolating neighborhood V ⊂ X. We first introduce Banach spaces of distributions supported on V, which are anisotropic versions of the usual space of C p functions C p (V) and of the generalized Sobolev spaces W p,t (V), respectively. Then we show that the transfer operators associated to T and a smooth weight g extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents. These bounds shed some light on those obtained by Kitaev for the radius of convergence of dynamical determinants.
Rigorous numerical investigation of the statistical properties of piecewise expanding maps  A feasibility study
, 2000
"... I explore the concrete applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps 1 Introduction The aim of the present paper is to investigate the possibility of answering questions of ..."
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Cited by 13 (1 self)
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I explore the concrete applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps 1 Introduction The aim of the present paper is to investigate the possibility of answering questions of the type: ffl Given a piecewise expanding map is it possible to decide if it is ergodic or mixing? ffl Is it possible to determine with arbitrary precision its absolutely continuous invariant measure? ffl If the map is mixing, is it possible to compute the exact rate of decay of correlations for a given function? Of course, the literature contains many papers in which some of these question are discussed either theoretically (especially, but not exclusively, as far as the invariant density is concerned) or numerically (e.g. [3], [4, 5, 6, 7], [8, 9], [14], [15], [18, 19], [21, 22], [23, 24, 25], [27, 28, 29, 30, 31, 32, 33], [34, 35], [38], [39, 40], [48], [49], [52], [55], [62], [66]). N...