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63
SRB measures for partially hyperbolic systems whose central direction is mostly expanding
, 2000
"... We construct SinaiRuelleBowen (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 nonuniformly expanding. If the r ..."
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Cited by 197 (44 self)
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We construct SinaiRuelleBowen (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 nonuniformly 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...
Large Deviations for Nonuniformly Hyperbolic Systems
, 2006
"... We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case ..."
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Cited by 70 (11 self)
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We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case of polynomial decay of correlations, we obtain polynomial large deviation estimates, and exhibit examples where these estimates are essentially optimal. In contrast with many treatments of large deviations, our methods do not rely on thermodynamic formalism. Hence, for Hölder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton, as well as polynomial estimates in situations where there is not a unique equilibrium measure. 1
The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems
, 2008
"... The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the BoltzmannGrad limit, where the radius of each scatterer tends to zero, and prove the ..."
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Cited by 45 (19 self)
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The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the BoltzmannGrad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.
A vectorvalued almost sure invariance principle for hyperbolic dynamical systems
, 2006
"... Abstract We prove an almost sure invariance principle (approximation by ddimensional Brownian motion) for vectorvalued H"older observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A diffeomorphisms and flows as well as systems modelled by ..."
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Cited by 44 (10 self)
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Abstract We prove an almost sure invariance principle (approximation by ddimensional Brownian motion) for vectorvalued H&quot;older observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A diffeomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2dimensional Brownian motion. 1 Introduction The scalar almost sure invariance principle (ASIP), or approximation by onedimensional Brownian motion, is a strong statistical property of sequences of random variables introduced by Strassen [40, 41]. It implies numerous other statistical limit laws including the central limit theorem, the functional central limit theorem, and the law of the iterated logarithm. See [23, 38] and references therein for a survey of consequences of the ASIP.
Central Limit Theorems and Invariance Principles for Lorenz Attractors
, 2006
"... We prove statistical limit laws for Hölder observations of the Lorenz attractor, and more generally for geometric Lorenz attractors. In particular, we prove the almost sure invariance principle (approximation by Brownian motion). Standard consequences of this result include the central limit theo ..."
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Cited by 40 (17 self)
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We prove statistical limit laws for Hölder observations of the Lorenz attractor, and more generally for geometric Lorenz attractors. In particular, we prove the almost sure invariance principle (approximation by Brownian motion). Standard consequences of this result include the central limit theorem, the law of the iterated logarithm, and the functional versions of these results.
Equilibrium states for interval maps: the potential −tlog Df
"... Abstract. We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials ϕ with the ‘bounded range ’ condition supϕ − inf ϕ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use ..."
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Cited by 36 (9 self)
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Abstract. We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials ϕ with the ‘bounded range ’ condition supϕ − inf ϕ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of PerronFrobenius operators. We demonstrate that this ‘bounded range ’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context. 1.
From rates of mixing to recurrence times via large deviations
 Adv. Math
"... Abstract. A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochasticlike behaviour such as large deviations or decay of correlations. Such geometric structures are generally hig ..."
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Cited by 22 (14 self)
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Abstract. A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochasticlike behaviour such as large deviations or decay of correlations. Such geometric structures are generally highly nontrivial and thus a natural question is the extent to which this approach can be applied. In this paper we show that in many cases stochasticlike behaviour itself implies that the system has certain nontrivial geometric properties, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration. As a by product of our techniques we also obtain some new results on large deviations for certain classes of systems which include Viana maps and multidimensional piecewise expanding maps.
Kinetic transport in the twodimensional periodic Lorentz gas
 NONLINEARITY 21 (2008), 1413–1422
, 2008
"... The periodic Lorentz gas describes an ensemble of noninteracting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (BoltzmannGrad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic trans ..."
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Cited by 20 (12 self)
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The periodic Lorentz gas describes an ensemble of noninteracting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (BoltzmannGrad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation—in contrast to the Lorentz gas with a disordered scatterer configuration. The present paper focuses on the twodimensional setup, and reports an explicit, elementary formula for the collision kernel of the transport equation.
Nice inducing schemes and the thermodynamics of rational maps
"... Abstract. We consider the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the (non)existence of equilibrium states of f for the potential −tln f ′ , and the analy ..."
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Cited by 20 (11 self)
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Abstract. We consider the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the (non)existence of equilibrium states of f for the potential −tln f ′ , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism of a rational map that is “expanding away from critical points ” and that has arbitrarily small “nice sets ” with some additional properties. Our results apply in particular to nonrenormalizable polynomials without indifferent periodic points, infinitely renormalizable quadratic polynomials with a priori bounds, real quadratic polynomials, topological ColletEckmann rational maps, and to backward contracting rational maps. As an application, for these maps we describe the dimension spectrum of Lyapunov exponents, and of pointwise dimensions of the measure of
Hyperbolic billiards and statistical physics, in
 International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich
"... Abstract. Mathematical theory of billiards is a fascinating subject providing a fertile source of new problems as well as conjecture testing in dynamics, geometry, mathematical physics and spectral theory. This survey is devoted to planar hyperbolic billiards with emphasis on their applications in ..."
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Cited by 19 (2 self)
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Abstract. Mathematical theory of billiards is a fascinating subject providing a fertile source of new problems as well as conjecture testing in dynamics, geometry, mathematical physics and spectral theory. This survey is devoted to planar hyperbolic billiards with emphasis on their applications in statistical physics, where they provide many physically interesting and mathematically tractable models. 1.