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70
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 and moderate deviations for slowly mixing dynamical systems
 Proc. Amer. Math. Soc
"... We obtain results on large deviations for a large class of nonuniformly hyperbolic dynamical systems with polynomial decay of correlations 1/n β, β> 0. This includes systems modelled by Young towers with polynomial tails, extending recent work of M. Nicol and the author which assumed β> 1. As ..."
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Cited by 23 (3 self)
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We obtain results on large deviations for a large class of nonuniformly hyperbolic dynamical systems with polynomial decay of correlations 1/n β, β> 0. This includes systems modelled by Young towers with polynomial tails, extending recent work of M. Nicol and the author which assumed β> 1. As a byproduct of the proof, we obtain slightly stronger results even when β> 1. The results are sharp in the sense that there exist examples (such as PomeauManneville intermittency maps) for which the obtained rates are best possible. In addition, we obtain results on moderate deviations. 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.
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
Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenzlike maps
"... Abstract. In this paper we establish extreme value statistics for observations on a class of hyperbolic systems: planar dispersing billiard maps and flows, Lozi maps and Lorenzlike maps. In particular we show that for time series arising from Hölder observations on these systems the successive m ..."
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Cited by 18 (5 self)
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Abstract. In this paper we establish extreme value statistics for observations on a class of hyperbolic systems: planar dispersing billiard maps and flows, Lozi maps and Lorenzlike maps. In particular we show that for time series arising from Hölder observations on these systems the successive maxima of the time series are distributed according to the corresponding extreme value distributions for independent identically distributed processes. These results imply an exponential law for the hitting and return time statistics of these dynamical systems. 1.
Optimal concentration inequalities for dynamical systems
 Commun. Math. Phys
, 2012
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STATISTICAL PROPERTIES OF ONEDIMENSIONAL MAPS UNDER WEAK HYPERBOLICITY ASSUMPTIONS
, 2011
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Natural equilibrium states for multimodal maps
 Comm. Math. Phys
"... Abstract. This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, ColletEckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials −tlog Df, for t ..."
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Cited by 11 (4 self)
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Abstract. This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, ColletEckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials −tlog Df, for the largest possible interval of parameters t. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained. 1.
Exponential decay of correlations in multidimensional dispersing billiards, preprint. 15
 Th. Dynam. Syst
"... decay of correlations in multidimensional dispersing billiards ..."
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Cited by 11 (0 self)
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decay of correlations in multidimensional dispersing billiards
Some unbounded functions of intermittent maps for which the central limit theorem holds
, 2007
"... Abstract. We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the PerronFrobenius operator of an expanding map T of [0,1] with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of f ◦T i, when f belo ..."
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Cited by 11 (6 self)
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Abstract. We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the PerronFrobenius operator of an expanding map T of [0,1] with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of f ◦T i, when f belongs to a large class of unbounded functions from [0,1] to R. We also prove other limit theorems and moment inequalities.