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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.
A vectorvalued almost sure invariance principle for Sinai billiards with random scatterers
, 2012
"... Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the to ..."
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Cited by 4 (2 self)
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Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of timedependent billiards on the one hand and in probability theory on the other, we prove a vectorvalued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly dependent and nonstationary. We provide an expression for the covariance matrix, which in the nonstationary case differs from the traditional one. We also obtain a new invariance principle for Sinai billiards (the case of fixed scatterers) with timedependent observables, and improve the accuracy and generality of existing results.
A functional analytic approach to perturbations of the Lorentz Gas. Comm. Math. Phys. (to appear). Stability of statistical properties in twodimensional piecewise hyperbolic maps
 Trans. Amer. Math. Soc
, 2008
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A note on operator semigroups associated to chaotic flows
"... Abstract. The transfer operator associated to a flow (continuous time dynamical system) is a oneparameter operator semigroup. We consider the operatorvalued Laplace transform of this oneparameter semigroup. Estimates on the Laplace transform have been used in various settings in order to show th ..."
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Cited by 1 (0 self)
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Abstract. The transfer operator associated to a flow (continuous time dynamical system) is a oneparameter operator semigroup. We consider the operatorvalued Laplace transform of this oneparameter semigroup. Estimates on the Laplace transform have been used in various settings in order to show the rate at which the flow mixes. Here we consider the case of exponential mixing and the case of rapid mixing (super polynomial). We develop the operator theory framework amenable to this setting and show that the same estimates may be used to produce results, in terms of the operators, which go beyond the results for the rate of mixing.
Electrical current in Sinai billiards under general small forces
 J. Stat. Phys
, 2013
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Exponential decay of correlations for finite horizon . . .
, 2014
"... We prove exponential decay of correlations for the billiard flow associated with a twodimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semigroup Lt of transfer operators, i.e., ..."
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We prove exponential decay of correlations for the billiard flow associated with a twodimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semigroup Lt of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions.
PSEUDOORBITS, STATIONARY MEASURES AND METASTABILITY
"... Abstract. We characterize absolutely continuous stationary measures (acsms) of randomly perturbed dynamical systems in terms of pseudoorbits linking the ergodic components of absolutely continuous invariant measures (acims) of the unperturbed system. We focus on those components, called leastelem ..."
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Abstract. We characterize absolutely continuous stationary measures (acsms) of randomly perturbed dynamical systems in terms of pseudoorbits linking the ergodic components of absolutely continuous invariant measures (acims) of the unperturbed system. We focus on those components, called leastelements, which attract pseudo orbits. Under the assumption that the transfer operators of both systems, the random and the unperturbed, satisfy a uniform LasotaYorke inequality on a suitable Banach space, we show that each least element is in a onetoone correspondence with an ergodic acsm of the random system. 1.