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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
Homoclinic and heteroclinic bifurcations in vector fields
 HANDBOOK OF DYNAMICAL SYSTEMS III, PP 379–524. ELSEVIER
, 2010
"... An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence o ..."
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Cited by 17 (0 self)
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An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multiround homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin’s method, Shil’nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.
Large deviations principles for nonuniformly hyperbolic rational maps
 Ann. Inst. H. Poincaré Anal. Non Linéaire
, 1998
"... Abstract. For a rational map satisfying the Topological ColletEckmann condition we prove a level2 large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that for such a rational map each Hölder continuous potential adm ..."
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Cited by 10 (2 self)
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Abstract. For a rational map satisfying the Topological ColletEckmann condition we prove a level2 large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that for such a rational map each Hölder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer. 1.
A LARGE DEVIATIONS BOUND FOR THE TEICHMÜLLER FLOW ON THE MODULI SPACE OF ABELIAN DIFFERENTIALS
, 2010
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LARGE DEVIATIONS BOUND FOR TEICHMÜLLER FLOW
, 2009
"... Large deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. The method is that of [1]. A corollary of the main results is a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, which extends earlier work ..."
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Large deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. The method is that of [1]. A corollary of the main results is a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, which extends earlier work of J. Athreya [2]. The “entropy approach” we use is similar to that of [20].
function, including the infinite horizon planar
, 2010
"... of correlations for flows with unbounded roof ..."
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Lorentzlike chaotic attractors revisited
, 2009
"... We describe some recent results on the dynamics of singularhyperbolic (Lorenzlike) attractors Λ introduced in [25]: (1) there exists an invariant foliation whose leaves are forward contracted by the flow; (2) there exists a positive Lyapunov exponent at every orbit; (3) attractors in this class a ..."
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We describe some recent results on the dynamics of singularhyperbolic (Lorenzlike) attractors Λ introduced in [25]: (1) there exists an invariant foliation whose leaves are forward contracted by the flow; (2) there exists a positive Lyapunov exponent at every orbit; (3) attractors in this class are expansive and so sensitive with respect to initial data; (4) they have zero volume if the flow is C², or else the flow is globally hyperbolic; (5) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the centerunstable Jacobian; (6) the hitting time associated to a geometric Lorenz attractor satisfies a logarithm law; (7) the rate of large deviations for the physical measure on the ergodic basin of a geometric Lorenz attractor is exponential.