Large deviations for nonuniformly hyperbolic systems

Melbourne, Ian; Nicol, Matthew

doi:10.1090/s0002-9947-08-04520-0

Cited by 161 publications

(257 citation statements)

References 38 publications

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“…Using the large deviation principle whereby deviations of time averages from the corresponding spatial averages are rare (Melbourne and Nicol, 2008), we argue…”

Section: Averaging In the Mori-zwanzig Frameworkmentioning

confidence: 98%

Stochastic Climate Theory

Gottwald¹,

Crommelin²,

Franzke³

2016

Nonlinear and Stochastic Climate Dynamics

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In this chapter we review stochastic modelling methods in climate science. First we provide a conceptual framework for stochastic modelling of deterministic dynamical systems based on the Mori-Zwanzig formalism. The Mori-Zwanzig equations contain a Markov term, a memory term and a term suggestive of stochastic noise. Within this framework we express standard model reduction methods such as averaging and homogenization which eliminate the memory term. We further discuss ways to deal with the memory term and how the type of noise depends on the underlying deterministic chaotic system. Secondly, we review current approaches in stochastic data-driven models. We discuss how the drift and diffusion coefficients of models in the form of stochastic differential equations can be estimated from observational data. We pay attention to situations where the data stems from multi scale systems, a relevant topic in the context of data from the climate system. Furthermore, we discuss the use of discrete stochastic processes (Markov chains) for e.g. stochastic subgrid-scale modeling and other topics in climate science.

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“…Using the large deviation principle whereby deviations of time averages from the corresponding spatial averages are rare (Melbourne and Nicol, 2008), we argue…”

Section: Averaging In the Mori-zwanzig Frameworkmentioning

confidence: 98%

Stochastic Climate Theory

Gottwald¹,

Crommelin²,

Franzke³

2016

Nonlinear and Stochastic Climate Dynamics

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“…It also follows that these equilibrium states have other statistical properties, such as the "almost sure invariant principle", see e.g. [Gou05,MN05,MN08,TK05].…”

Section: Analyticity Of the Pressure Function: The Pressure Function mentioning

confidence: 99%

Nice Inducing Schemes and the Thermodynamics of Rational Maps

Przytycki

Rivera-Letelier

2010

Commun. Math. Phys.

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Abstract:We study 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 existence of equilibrium states of f for the potential −t ln f , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in R, with at most two exceptions.

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“…As observed in [28], it is then possible to use an inequality due to Rio [35, Theorem 2.5] (see [30,Proposition 7]) to obtain an improved result on the order of converging moments.…”

Section: Convergence Of Moments For Mapsmentioning

confidence: 99%

Convergence of moments for Axiom A and non-uniformly hyperbolic flows

Melbourne

Török

2011

Ergod. Th. Dynam. Sys.

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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 Hénon-like attractors, Lorenz attractors, semidispersing billiards, finite horizon planar periodic Lorentz gases, and Pomeau-Manneville intermittency maps.

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Large deviations for nonuniformly hyperbolic systems

Cited by 161 publications

References 38 publications

Stochastic Climate Theory

Stochastic Climate Theory

Nice Inducing Schemes and the Thermodynamics of Rational Maps

Convergence of moments for Axiom A and non-uniformly hyperbolic flows

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