Spatial structure of Sinai–Ruelle–Bowen measures

Chernov, N.; Korepanov, Alexey

doi:10.1016/j.physd.2014.06.006

Cited by 5 publications

(8 citation statements)

References 18 publications

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“…We note that the stronger linear response property can be established in certain situations [16], but that linear response is not required for the purposes of this paper.…”

Section: Nonuniformly Hyperbolic Fast Dynamicsmentioning

confidence: 99%

Martingale–coboundary decomposition for families of dynamical systems

Korepanov

Kosloff

Melbourne

2018

Ann. Inst. H. Poincaré Anal. Non Linéaire

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A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. AbstractWe prove statistical limit laws for sequences of Birkhoff sums of the typewhere T n is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family T n is replaced by a fixed transformation T , and which is particularly effective in the case when T n varies with n.In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family T n consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards.As an application, we prove a homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

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“…We note that the stronger linear response property can be established in certain situations [16], but that linear response is not required for the purposes of this paper.…”

Section: Nonuniformly Hyperbolic Fast Dynamicsmentioning

confidence: 99%

Martingale–coboundary decomposition for families of dynamical systems

Korepanov

Kosloff

Melbourne

2018

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…There is no need to introduce the homogeneous strips if lim x→0 + T (x) < ∞. In the case when lim x→0 + T (x) = ∞, the function f (x, y) defined in (16) vanishes at x = 0. If Condition (h2) also holds, we choose a constant b > 0 such that…”

Section: Verification Of Assumptions (H1) -(H4)mentioning

confidence: 99%

“…Recall that S 1 consists of vertical singularity lines and ∂Q, and thus d(x, S 1 ) min{x, y, 1 − y} for x = (x, y) with x ∈ I 00 . By the definition of f (x) in (16), we obtain that f (x) = f (x) = 1…”

Section: Verification Of Assumptions (H1) -(H4)mentioning

confidence: 99%

“…provided that k 0 is sufficiently large. [16] (see also Appendix A in [11]), we can project an SRB measure for F : Q → Q along the vertical stable foliation down to an absolutely continuous invariant probability measure µ, from which the statistical properties of the one-dimensional system (F, µ), such as the exponential decay of correlations, the large deviation principle and the almost-sure invariance principle follow immediately.…”

Section: 13mentioning

confidence: 99%

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Statistical properties of one-dimensional expanding maps with singularities of low regularity

Chen¹,

Zhang²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We investigate the statistical properties of piecewise expanding maps on the unit interval, whose inverse Jacobian may have low regularity near singularities. The method is new yet simple: instead of directly working with the 1-d map, we first lift the 1-d expanding map to a hyperbolic map on the unit square, and then take advantage of the functional analytic method developed by Demers and Zhang in [21,22,23] for hyperbolic systems with singularities. By projecting back to the 1-d map, we are able to prove that it inherits nice statistical properties, including the large deviation principle, the exponential decay of correlations, as well as the almost sure invariance principle for the expanding map on a large class of observables. Moreover, we are able to prove that the projected SRB measure has a piecewise continuous density function. Our results apply to rather general 1-d expanding maps, including some C 1 perturbations of the Lorenz-like map and the Gauss map whose statistical properties are still unknown as they fail all other available methods.2010 Mathematics Subject Classification. Primary: 37D50, 37A25.

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“…The Green-Kubo type formula was proved, and it was shown that the current generated by the forced flow is closely related to the strength of the force. Very recently, Chernov and Korepanov investigated in [CK13] the linear response for Sinai billiards under external forces (without twisting). In this paper, we consider the dynamics of the Sinai billiard on the table Q, but subject to more general forces P = (F, G) both during flight and at collisions.…”

Section: Introductionmentioning

confidence: 99%

Electrical Current in Sinai Billiards Under General Small Forces

Chernov

Zhang

2013

J Stat Phys

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The Lorentz gas of Z 2 -periodic scatterers (or the so called Sinai billiards) can be used to model motion of electrons on an ionized medal. We investigate the linear response for the system under various external forces (during both the flight and the collision). We give some characterizations under which the forced system is time-reversible, and derive an estimate of the electrical current generated by the forced system. Moreover, applying Pesin entropy formula and Young dimension formula, we get several characterizations of the non-equilibrium steady state of the forced system.

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Spatial structure of Sinai–Ruelle–Bowen measures

Cited by 5 publications

References 18 publications

Martingale–coboundary decomposition for families of dynamical systems

Martingale–coboundary decomposition for families of dynamical systems

Statistical properties of one-dimensional expanding maps with singularities of low regularity

Electrical Current in Sinai Billiards Under General Small Forces

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