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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
Decay of Correlations for Slowly Mixing Flows
, 2006
"... We show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of a class of nonuniformly hyperbolic diffeomorphisms for which Young proved polynomial decay of correlations. Roughly speaking, in situations where t ..."
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Cited by 17 (10 self)
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We show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of a class of nonuniformly hyperbolic diffeomorphisms for which Young proved polynomial decay of correlations. Roughly speaking, in situations where the decay rate O(1/n β) has previously been proved for diffeomorphisms, we establish the decay rate O(1/t β) for typical flows. Applications include certain classes of semidispersing billiards, as well as dispersing billiards with vanishing curvature. In addition, we obtain results for suspension flows with unbounded roof functions. This includes the planar periodic Lorentz flow with infinite horizon. 1
Convergence of moments for Axiom A and nonuniformly hyperbolic flows
, 2010
"... In the paper, we prove convergence of moments of all orders for Axiom A diffeomorphisms and flows. The same results hold for nonuniformly hyperbolic diffeomorphisms and flows modelled by Young towers with superpolynomial tails. For polynomial tails, we prove convergence of moments up to a certain or ..."
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Cited by 5 (2 self)
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In the paper, we prove convergence of moments of all orders for Axiom A diffeomorphisms and flows. The same results hold for nonuniformly hyperbolic diffeomorphisms and flows modelled by Young towers with superpolynomial tails. For polynomial tails, we prove convergence of moments up to a certain order, and give examples where moments diverge when this order is exceeded. Nonuniformly hyperbolic systems covered by our result include Hénonlike attractors, Lorenz attractors, semidispersing billiards, finite horizon planar periodic Lorentz gases, and PomeauManneville intermittency maps. 1
MIXING RATE FOR SEMIDISPERSING BILLIARDS WITH NONCOMPACT CUSPS
"... Abstract. Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even polynomial decay. However, until now nothing is known about ..."
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Abstract. Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even polynomial decay. However, until now nothing is known about mixing properties for billiard tables with noncompact cusps. There is no consensual definition of mixing for systems with infinite invariant measure. In this paper we study geometric and ergodic properties of billiard tables with a noncompact cusp. The goal of this text is, using the definition of mixing proposed by Krengel and Sucheston for systems with invariant infinite measure, to show that the billiard whose table is constituted by the xaxis and and the portion in the plane below the graph of f(x) = 1 x+1 is mixing and the speed of mixing is polynomial. 1.
International Journal of Bifurcation and Chaos c © World Scientific Publishing Company
, 2013
"... Polynomial decay of correlations in the generalized baker’s transformation ..."
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Polynomial decay of correlations in the generalized baker’s transformation
and
, 2008
"... of correlations and invariance principles for dispersing billiards with cusps, ..."
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of correlations and invariance principles for dispersing billiards with cusps,
Supported by the Austrian Federal Ministry of Education, Science and Culture
"... Let (X,T,µ) be a dynamical system, i.e. T: X → X is a map preserving a probability measure µ. The dynamics is seen as a sequence x0,x1,...,xn,... of points in X such that xn = T n (x0). Alternatively, we may only ‘see ’ the values F0,F1,...,Fn,... of some function F: X → R, where Fn = F(xn). We call ..."
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Let (X,T,µ) be a dynamical system, i.e. T: X → X is a map preserving a probability measure µ. The dynamics is seen as a sequence x0,x1,...,xn,... of points in X such that xn = T n (x0). Alternatively, we may only ‘see ’ the values F0,F1,...,Fn,... of some function F: X → R, where Fn = F(xn). We call F an observable. If the sequence {Xn}, or {Fn}, was independent (relative to the measure µ), then we could easily apply all major results of classical probability theory... But this is almost never the case in deterministic systems. So let us assume the simplest situation with dependence: the function F only takes finitely many values, say {1, 2,...,I}, and the sequence {Fn} is a Markov chain; i.e. Fn depends only on Fn−1 but not on the previous values Fn−m, m ≥ 2. This Markov chain has a stationary distribution P with components pi = µ(F0 = i) and its transition probability matrix Π has components πij = µ(F1 = j/F0 = i). If πij = pj for all i,j we would have an independent sequence. Suppose πij ≥ γpj for some γ> 0 (the components of the matrix Πn) converge to the stationary distribution exponentially fast in the following sense: and all i,j. Then the nstep transition probabilities π (n) ij (1) Var(Π (n) i,P) ≤ (1 − γ) n. = (π(n) i1,...,π(n) iI) is the ‘image ’ of the ith state at time n and P = (p1,...,pI) is the stationary vector; we denote by Var the distance in variation between probability vectors:
Fluctuations of observables in dynamical
"... systems: from limit theorems to concentration inequalities JeanRene ́ Chazottes Abstract We start by reviewing recent probabilistic results on ergodic sums in a large class of (nonuniformly) hyperbolic dynamical systems. Namely, we describe the central limit theorem, the almostsure convergence to ..."
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systems: from limit theorems to concentration inequalities JeanRene ́ Chazottes Abstract We start by reviewing recent probabilistic results on ergodic sums in a large class of (nonuniformly) hyperbolic dynamical systems. Namely, we describe the central limit theorem, the almostsure convergence to the Gaussian and other stable laws, and large deviations. Next, we describe a new branch in the study of probabilistic properties of dynamical systems, namely concentration inequalities. They allow to describe the fluctuations of very general observables and to get bounds rather than limit laws. We end up with two sections: one gathering various open problems, notably on random dynamical systems, coupled map lattices and socalled nonconventional ergodic averages; and another one giving pointers to the literature about moderate