Transfer Operators and Limit Laws for Typical Cocycles

Park, Kiho; Piraino, Mark

doi:10.1007/s00220-021-04300-x

Cited by 8 publications

(5 citation statements)

References 31 publications

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“…Note that compared with the main result in [27], the positivity of the variance is an implicit conclusion rather than an additional hypothesis.…”

Section: Introductionmentioning

confidence: 68%

“…For the case of linear cocycles over hyperbolic systems, in the same setting of this paper, S. Gouëzel and L. Stoyanov [20] obtained a large deviations principle, while K. Park and M. Piraino [27] obtained a central limit theorem (and a large deviations principle).…”

Section: Introductionmentioning

confidence: 71%

“…In other related works, e.g. [27], it is the transfer operator that plays a similar rôle. Our method allows for a more quantitative control of various parameters, thus ensuring the uniformity of the LDT estimate above.…”

Section: Introductionmentioning

confidence: 99%

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Hölder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps

Duarte¹,

Klein²,

Poletti³

2021

Preprint

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Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a central limit theorem and a large deviations principle.

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“…Note that compared with the main result in [27], the positivity of the variance is an implicit conclusion rather than an additional hypothesis.…”

Section: Introductionmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 71%

See 1 more Smart Citation

Hölder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps

Duarte¹,

Klein²,

Poletti³

2021

Preprint

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“…This fact is well known in our setup it can be deduced in the same way as [12]. Notice that in order to ensure that ρ(L β ) is differentiable we need that T is irreducible.…”

Section: The Determinant and A Formula For The Lyapunov Exponentmentioning

confidence: 74%

Fast approximation of Lyapunov exponents: Beyond the locally constant case

Piraino¹

2021

Preprint

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We study the problem of estimating the maximal Lyapunov exponent of dominated cococycles. In particular we are concerned with cocycles over Gibbs states on shifts of finite type for which both the function defining the cocycle and the potential defining the Gibbs state may depend on infinitely many coordinates but are still very regular. We show that when the nth variation of both the cocycle and the potential is O(e −cn 2 ) for some c > h top then using periodic points of period less then n the Lyapunov exponent can be approximated to an accuracy O(n −kn ) for some explicit k > 0.

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“…Typicality was first introduced by Bonatti and Viana [BV04] to establish the simplicity of the Lyapunov exponents for any invariant measure with continuous local product structure. Later, it was shown that typical cocycles possess other interesting properties, including the uniqueness of the equilibrium states for the singular value potentials [Par20] and the validity of various limit laws [DKP22, PP22]. The following theorem is another interesting property that can be derived from the typicality assumption.…”

Section: Introductionmentioning

confidence: 99%