Lagrange approximation of transfer operators associated with holomorphic data

Bandtlow, Oscar F.; Slipantschuk, Julia

doi:10.48550/arxiv.2004.03534

Cited by 7 publications

(23 citation statements)

References 30 publications

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“…Now, because of its definition through function composition (5), the transfer operator L α is bounded from larger Hardy spaces into smaller ones [4].…”

Section: As An Edge Case Thementioning

confidence: 99%

“…The order of this approximation (i.e. the number of Chebyshev coefficients used) was chosen to be approximately that used by Poltergeist.jl for estimating the right eigenfunctions 4…”

Section: Local Manifold Approximationsmentioning

confidence: 99%

“…This implies that the so-called Chebyshev Galerkin approximation P n L α of the transfer operator L α converges exponentially to the true operator, and thus so do its spectrum and eigenvalues [4]. This can also be extended to more complex functions of L α such as the differential of F t defined in (9) below.…”

mentioning

confidence: 94%

“…Our evidence for the homoclinic tangency is numerical. To approximate the infinite-dimensional limit system, we will apply Chebyshev Galerkin discretisations for the transfer operators that describe the thermodynamic limit [45,4], using the software package Poltergeist.jl [44]. Such discretisations are very accurate and efficient in approximating such objects.…”

Section: Introductionmentioning

confidence: 99%

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Non-hyperbolicity at large scales of a high-dimensional chaotic system

Wormell¹

2021

Preprint

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Many important high-dimensional dynamical systems exhibit complex chaotic behaviour. Their complexity means that their dynamics are necessarily comprehended under strong reducing assumptions. It is therefore important to have a clear picture of these reducing assumptions' range of validity. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively hyperbolic, which implies many felicitous statistical properties. We demonstrate, contrary to the chaotic hypothesis, the existence of non-hyperbolic large-scale dynamics in a mean-field coupled system. To do this we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev Galerkin transfer operator discretisation. This enables us to obtain a high precision estimate of a homoclinic tangency, implying a failure of hyperbolicity. Robust nonhyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be assumed to hold in all systems, and a better understanding of the domain of its validity is required.

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“…Now, because of its definition through function composition (5), the transfer operator L α is bounded from larger Hardy spaces into smaller ones [4].…”

Section: As An Edge Case Thementioning

confidence: 99%

“…The order of this approximation (i.e. the number of Chebyshev coefficients used) was chosen to be approximately that used by Poltergeist.jl for estimating the right eigenfunctions 4…”

Section: Local Manifold Approximationsmentioning

confidence: 99%

mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Non-hyperbolicity at large scales of a high-dimensional chaotic system

Wormell¹

2021

Preprint

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“…The induced map is uniformly expanding (see Figure 2), and it is therefore possible to apply results on uniformly expanding dynamics to it, as well as various numerical methods [18,5,4]. The non-mixing dynamics near the fixed point poses a problem for obtaining accurate numerical estimates for these maps.…”

Section: Introductionmentioning

confidence: 99%

Efficient computation of statistical properties of intermittent dynamics

Wormell¹

2021

Preprint

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Intermittent maps of the interval are simple and widely-studied models for chaos with slow mixing rates, but have been notoriously resistant to numerical study. In this paper we present an effective framework to compute many ergodic properties of these systems, in particular invariant measures and mean return times. The framework combines three ingredients that each harness the smooth structure of these systems' induced maps: Abel functions to compute the action of the induced maps, Euler-Maclaurin summation to compute the pointwise action of their transfer operators, and Chebyshev Galerkin discretisations to compute the spectral data of the transfer operators. The combination of these techniques allows one to obtain exponential convergence of estimates for polynomially growing computational outlay, independent of the order of the map's neutral fixed point. This enables effective numerical exploration of intermittent dynamics in all parameter regimes, including in the infinite ergodic regime.

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Effective estimates of Lyapunov exponents for random products of positive matrices

Pollicott

2021

Nonlinearity

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In this note we describe estimates on the error when calculating the Lyaponov exponent for random products of positive matrices using dynamical determinants. This extends the results in (Jurga N and Morris I 2019 Nonlinearity 32 4117–46; Pollicott M 2010 Invent. Math. 181 209–26) by drawing upon a new approach introduced in (Jenkinson O and Pollicott M 2018 Adv. Math. 325 87–115; Jenkinson O, Pollicott M and Vytnova P 2018 J. Stat. Phys. 170 221–53).

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Lagrange approximation of transfer operators associated with holomorphic data

Cited by 7 publications

References 30 publications

Non-hyperbolicity at large scales of a high-dimensional chaotic system

Non-hyperbolicity at large scales of a high-dimensional chaotic system

Efficient computation of statistical properties of intermittent dynamics

Effective estimates of Lyapunov exponents for random products of positive matrices

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