Almost sure rates of mixing for partially hyperbolic attractors

Alves, José F.; Bahsoun, Wael; Ruziboev, Marks

doi:10.1016/j.jde.2021.12.008

Cited by 11 publications

(30 citation statements)

References 54 publications

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“…unimodel maps admit a random tower extension, and obtained almost sure rates of mixing (decay of correlations). Results in this directions were also obtained later by several authors [1,6,7,8,19]. In [41] the author proved an almost sure invarinace principle (ASIP) for random Young towers.…”

Section: Introductionmentioning

confidence: 57%

“…Therefore, there is a constant c > 0 so that, P -a.s. when n is large enough we can partition L it,n ω into at least cn blocks so that the norm of the odd blocks does not exceed B J , while the norm of the even blocks does not exceed 1 2 B J (we can take c = P (V )/2s). Therefore, P -a.s. for any n large enough we have…”

Section: Assumption (I)mentioning

confidence: 99%

“…The first four assumptions are straight forward generalizations of classical assumptions, and they mean the maps f ω are a random family of non-uniformly distance expanding maps, while the sixth assumption comes from [7] (see also [1] and [19]). Under these assumptions, the map π ω : ∆ ω → M ω given by π ω (x, ℓ) = f ℓ σ −ℓ ω ω is a Holder continuous bijection on its image.…”

Section: Assumption (I)mentioning

confidence: 99%

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Limit theorems for random non-uniformly expanding or hyperbolic maps with exponential tails

Hafouta¹

2020

Preprint

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We prove a Berry-Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random tower extension in the sense of Young, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.

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Section: Introductionmentioning

confidence: 57%

Section: Assumption (I)mentioning

confidence: 99%

Section: Assumption (I)mentioning

confidence: 99%

See 1 more Smart Citation

Limit theorems for random non-uniformly expanding or hyperbolic maps with exponential tails

Hafouta¹

2020

Preprint

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“…We emphasize that, compared with the precise class of expanding on average random dynamical systems introduced in [12], we only need the additional mild assumption that the map ω → T ω has a countable range: this is to ensure that we can apply a suitable version of the multiplicative ergodic theorem [24]. 2 We note that there are previous works devoted to limit theorems for random dynamical system that allow contracting behavior on large measure sets [1], but only under the condition that the family (T ω ) ω∈Ω only takes finitely many values (and assuming that (Ω, F , P, σ) is a Bernoulli shift),or don't require the presence of a uniform decay of correlations, such as the work of Kifer [39] (partially inspired by the work of Cogburn [14]) as well as the first author and Hafouta [21]. Roughly speaking, the main idea in those papers is to pass to the associated induced system, where the inducing is done with respect to the region of Ω on which one has the uniform decay of correlations.…”

Section: Main Contributions Of the Present Papermentioning

confidence: 99%

“…Indeed, those are key tools to model many natural phenomena, including the transport in complex environments such as in the ocean or the atmosphere [3]: it is therefore crucial to understand their long term quantitative behavior. Among many remarkable contributions, we particularly emphasize those dealing with the decay of correlations [2,6,7,9,12,15], various (quenched or annealed) limit laws [1,4,16,17,18,19,20,30,31,32,39,43,44,48], as well as recent results devoted to the linear response of random dynamical systems [8,23,47]. For similar results in the closely related context of sequential dynamical systems, we refer to [10,11,33,35,36,37,29,42] and references therein.…”

Section: Introductionmentioning

confidence: 98%

Quenched limit theorems for expanding on average cocycles

Dragičević¹,

Sedro²

2021

Preprint

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We prove quenched versions of a central limit theorem, a large deviations principle as well as a local central limit theorem for expanding on average cocycles. This is achieved by building an appropriate modification of the spectral method for nonautonomous dynamics developed by Dragičević et al. (Comm Math Phys 360: 1121-1187, to deal with the case of random dynamics that exhibits nonuniform decay of correlations, which are ubiquitous in the context of the multiplicative ergodic theory. Our results provide an affirmative answer to a question posed by Buzzi (Comm Math Phys 208: 25-54, 1999).

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Limit Theorems for Random Non-uniformly Expanding or Hyperbolic Maps with Exponential Tails

Hafouta

2021

Ann. Henri Poincaré

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We prove central limit theorems and related results for non-uniformly expanding random dynamical systems and general random Gibbs measures under some integrability conditions and some (relatively weak) mixing related assumptions on the random environment/base map. Some of our conditions also apply to maps which only expand on average, in an appropriate sense. This is the first set of conditions that allows to treat general non-uniformly expanding cases. A crucial step in our proofs can be viewed as effective moderate versions of the random Ruelle-Perron-Frobenius (RPF) theorem for the random transfer operator cocycle corresponding to the random Gibbs measures (namely, effective moderate estimates on the rate of "convergence" to the projection on the top Lyapunov space in the corresponding multiplicative ergodic theorem). As a byproduct of our methods we are also able to prove statistical properties of the corresponding skew products and some estimates on the tails of random mixing times. Other applications like stochastic homogenization and effective linear response formulas will be considered in future projects. Compared with recent progress [40] on the subject, we are able to treat random Gibbs measures corresponding to potentials which do not necessarily have small local oscillation along inverse branches. Moreover, we are able to consider more general classes of random expanding maps with unbounded mixing/covering times and (only) local pairing property. Thus, our results apply to a wide family of C 2 -random expanding maps on compact C 2 manifolds, together with the random geometric potential which corresponds to the unique equivariant family of measures which are absolutely continuous with respect to the volume measure. Some of our results are also new in the case of small local oscillation. Other applications include high dimensional piecewise expanding maps and random subshifts of finite type, which have applications to random Markov interval maps and finite state Markov chains in random dynamical environment.

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Almost sure rates of mixing for partially hyperbolic attractors

Cited by 11 publications

References 54 publications

Limit theorems for random non-uniformly expanding or hyperbolic maps with exponential tails

Limit theorems for random non-uniformly expanding or hyperbolic maps with exponential tails

Quenched limit theorems for expanding on average cocycles

Limit Theorems for Random Non-uniformly Expanding or Hyperbolic Maps with Exponential Tails

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