Mixing rates for intermittent maps of high exponent

Terhesiu, Dalia

doi:10.1007/s00440-015-0690-0

Cited by 13 publications

(11 citation statements)

References 28 publications

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“…To be more precise, when α ≥ 1, each intermittent map admits a unique (up to scaling) σ-finite (but not finite), absolutely continuous invariant measure µ. In these infinite measure settings, although one may still obtain certain mixing rate [0,1] f g • T n α dµ for some reasonably well-behaved observables f, g. (See e.g., [16,22] for the concrete statements), there is no analogous result as in (9) to the best of our knowledge. Secondly, the classical Borel-Cantelli lemma does not hold in the infinite measure setting.…”

Section: Discussion For Further Researchmentioning

confidence: 99%

A note on the run length function for intermittent maps

Cui

Fang

Zhang

2019

Journal of Mathematical Analysis and Applications

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We study the run length function for intermittent maps. In particular, we show that the longest consecutive zero digits (resp. one digits) having a time window of polynomial (resp. logarithmic) length. Our proof is relatively elementary in the sense that it only relies on the classical Borel-Cantelli lemma and the polynomial decay of intermittent maps. Our results are compensational to the Erdős-Rényi law obtained by Denker and Nicol in [8]. * Corresponding author, and every author contributes equally.

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Section: Discussion For Further Researchmentioning

confidence: 99%

A note on the run length function for intermittent maps

Cui

Fang

Zhang

2019

Journal of Mathematical Analysis and Applications

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“…Our results on mixing when β ∈ ( 1 2 , 1] for semiflows (Corollary 3.1 and the extensions in Section 9) and for flows (Theorem 10.5), are proved by adapting Erickson's methods to the deterministic setting.For β ≤ 1 2 , additional hypotheses are needed on the tail of τ to obtain a strong renewal theorem (and hence mixing) even for discrete time; see [15,19,22] for i.i.d. results and [24] for deterministic results (see also [41] for higher order theory in both the i.i.d. and deterministic settings).…”

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confidence: 81%

Renewal theorems and mixing for non Markov flows with infinite measure

Melbourne¹,

Terhesiu²

2020

Ann. Inst. H. Poincaré Probab. Statist.

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We obtain results on mixing for a large class of (not necessarily Markov) infinite measure semiflows and flows. Erickson proved, amongst other things, a strong renewal theorem in the corresponding i.i.d. setting. Using operator renewal theory, we extend Erickson's methods to the deterministic (i.e. noni.i.d.) continuous time setting and obtain results on mixing as a consequence.Our results apply to intermittent semiflows and flows of Pomeau-Manneville type (both Markov and nonMarkov), and to semiflows and flows over Collet-Eckmann maps with nonintegrable roof function.Here, ℓ : [0, ∞) → [0, ∞) is a measurable slowly varying function (so lim t→∞ ℓ(λt)/ℓ(t) = 1 for all λ > 0). Consider the suspension (Y τ , µ τ ) and suspension semiflow F t : Y τ → Y τ (the standard definitions are recalled in Section 3).The aim is to prove a mixing result of the formfor a suitable normalisation a t → ∞ and suitable classes of observables v, w :Under certain hypotheses, [13,37] obtained results on mixing and rates of mixing for such semiflows. The hypotheses were of two types: (i) assumptions on "renewal operators" associated to the transfer operator of F and the roof function τ , and (ii) Dolgopyat-type assumptions of the type used to obtain mixing rates for finite measure (semi)flows [17].As pointed out to us by Dima Dolgopyat, Péter Nándori and Doma Szász, mixing for indicator functions can be regarded as a local limit theorem and hence hypotheses of type (ii) should not be necessary.In this paper, we show that operator renewal-theoretic assumptions (i) are indeed sufficient for obtaining the mixing results in [13,37]. The abstract framework in [13] turns out again to be flexible enough to cover nonMarkov situations. Moreover, our main results extend to flows and we are able to treat large classes of observables v, w. (Conditions of type (i) alone are not sufficient for obtaining rates of mixing; the best results remain those in [13].)The analogous probabilistic results go back to Erickson [20] who obtained strong renewal theorems in an i.i.d. continuous time framework under the assumption β ∈ ( 1 2 , 1]. (In the discrete time setting, see [22] for the i.i.d. case and [35] for the deterministic case.) Our results on mixing when β ∈ ( 1 2 , 1] for semiflows (Corollary 3.1 and the extensions in Section 9) and for flows (Theorem 10.5), are proved by adapting Erickson's methods to the deterministic setting.For β ≤ 1 2 , additional hypotheses are needed on the tail of τ to obtain a strong renewal theorem (and hence mixing) even for discrete time; see [15,19,22] for i.i.d. results and [24] for deterministic results (see also [41] for higher order theory in both the i.i.d. and deterministic settings). For the continuous time case, Dolgopyat & Nándori [18] obtain strong renewal theorems for a class of Markov semiflows including the range β ≤ 1 2 (again under extra hypotheses on the tail µ(τ > t)), though our main examples seem beyond their framework. In the absence of additional tail hypotheses, [20] showed how to obtain a partial resul...

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“…See [27,36] and also [22,33,34] for improved asymptotics. (c) One obtains a limit theorem (stable law) via the spectral method, the Nagaev-Guivarc'h (also referred to as Aaronson-Denker) method, for f τ ; we refer to the comprehensive survey [20] and to the original papers [2,21].…”

Section: Methods Of Proof and The Main Dynamical Ingredientmentioning

confidence: 99%

Limit theorems for wobbly interval intermittent maps

2021

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We consider perturbations of interval maps with indifferent fixed points, which we refer to as wobbly interval intermittent maps, for which stable laws for general Hölder observables fail. We obtain limit laws for such maps and Hölder observables. These limit laws are similar to the classical semistable laws previously established for random processes. One of the examples considered is an interval map with a countable number of discontinuities, and to analyse it we need to construct a Markov/Young tower.

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Mixing rates for intermittent maps of high exponent

Cited by 13 publications

References 28 publications

A note on the run length function for intermittent maps

A note on the run length function for intermittent maps

Renewal theorems and mixing for non Markov flows with infinite measure

Limit theorems for wobbly interval intermittent maps

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