Results 1  10
of
27
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.
Global Attractors in Partial Differential Equations
"... this paper, we present the weakly damped Schrodinger equation, which is a system generated by a dispersive equation with weak damping. ..."
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Cited by 18 (0 self)
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this paper, we present the weakly damped Schrodinger equation, which is a system generated by a dispersive equation with weak damping.
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 15 (7 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
Comparison Theorems And Orbit Counting In Hyperbolic Geometry
 Trans. Amer. Math. Soc
, 1998
"... In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ..."
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Cited by 11 (11 self)
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In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both "thermodynamic" ergodic theory and the automaton associated to strongly Markov groups. 0.
OrnsteinZernike Theory for the Finite Range Ising Models above ...
, 2001
"... We derive precise OrnsteinZernike asymptotic formula for the decay of the twopoint function (rr0rr)z in the general context of finite range Ising type models on Z a. ..."
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Cited by 9 (3 self)
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We derive precise OrnsteinZernike asymptotic formula for the decay of the twopoint function (rr0rr)z in the general context of finite range Ising type models on Z a.
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 brea ..."
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Cited by 8 (1 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.
Transfer operators acting on Zygmund functions
 Trans. Amer. Math. Soc
, 1995
"... . We obtain a formula for the essential spectral radius ae ess of transfertype operators associated with families of C 1+ffi diffeomorphisms of the line and Zygmund, or Holder, weights acting on Banach spaces of Zygmund (respectively Holder) functions. In the uniformly contracting case the essent ..."
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Cited by 6 (4 self)
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. We obtain a formula for the essential spectral radius ae ess of transfertype operators associated with families of C 1+ffi diffeomorphisms of the line and Zygmund, or Holder, weights acting on Banach spaces of Zygmund (respectively Holder) functions. In the uniformly contracting case the essential spectral radius is strictly smaller than the spectral radius when the weights are positive. 1. Introduction During the last decade, a generalised theory of Fredholm determinants has been obtained using tools from statistical mechanics, often in a dynamical setting. Typically, one considers  a transformation f , with finitely or countably many inverse branches, of a metric space M to itself,  a weight g : M ! C ; and one defines the associated transfer operator L'(z) = X f(w)=z g(w)'(w) acting on a Banach space of functions ' : M ! C . Transfer operators are useful in the study of "interesting" invariant measures for f . They sometimes arise in a surprising fashion: It has been prov...
The essential spectrum of advective equations
"... Abstract. The geometric optics stability method is extended to a general class of linear advective PDE’s with pseudodifferential bounded perturbation. We give a new short proof of Vishik’s formula for the essential spectral radius. We show that every point in the dynamical spectrum of the correspond ..."
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Cited by 5 (1 self)
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Abstract. The geometric optics stability method is extended to a general class of linear advective PDE’s with pseudodifferential bounded perturbation. We give a new short proof of Vishik’s formula for the essential spectral radius. We show that every point in the dynamical spectrum of the corresponding bicharacteristicamplitude system contributes a point into the essential spectrum of the PDE. Generic spectral pictures are obtained in Sobolev spaces of sufficiently large smoothness. Applications to instability are presented.
Quasicompactness of transfer operators for contact anosov flows, Nonlinearity 23
, 2010
"... Abstract. For any C r contact Anosov flow with r ≥ 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all C r functions, such that the transfer operators for the flow extend to them boundedly and that the extensions are quasico ..."
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Cited by 5 (1 self)
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Abstract. For any C r contact Anosov flow with r ≥ 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all C r functions, such that the transfer operators for the flow extend to them boundedly and that the extensions are quasicompact. Further we give explicit bounds on the essential spectral radii of the extensions in terms of the differentiability r and the hyperbolicity exponents of the flow. 1.
On systems with finite ergodic degree
 Far East J. Dynam. Syst
"... In this paper we study the ergodic theory of a class of symbolic dynamical systems (Ω,T,µ) where T: Ω → Ω the left shift transformation on Ω = ∏ ∞ 0 {0,1} and µ is a σfinite Tinvariant measure having the property that one can find a real number d so that µ(τ d) = ∞ but µ(τ d−ǫ) < ∞ for all ǫ> 0 ..."
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Cited by 4 (0 self)
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In this paper we study the ergodic theory of a class of symbolic dynamical systems (Ω,T,µ) where T: Ω → Ω the left shift transformation on Ω = ∏ ∞ 0 {0,1} and µ is a σfinite Tinvariant measure having the property that one can find a real number d so that µ(τ d) = ∞ but µ(τ d−ǫ) < ∞ for all ǫ> 0, where τ is the first passage time function in the reference state 1. In particular we shall consider invariant measures µ arising from a potential V which is uniformly continuous but not of summable variation. If d> 0 then µ can be normalized to give the unique nonatomic equilibrium probability measure of V for which we compute the (asymptotically) exact mixing rate, of order n −d. We also establish the weakBernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead d ≤ 0 then µ is an infinite measure with scaling rate of order n d. Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operatorvalued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their nonpolar singularity at z = 1 is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point. 1