On stochastic stability of non-uniformly expanding interval maps

Shen, Weixiao

doi:10.1112/plms/pdt013

Cited by 16 publications

(23 citation statements)

References 37 publications

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“…We would like also to suggest here a weakening of the linear response problem: Consider a one-parameter family f t of (say, smooth unimodal maps) through f t0 and, for each ǫ > 0, a random perturbation of f t with unique invariant measure µ ǫ t , e.g., like in [39]. Then for each positive ǫ, it should not be very difficult to see that the map t → µ ǫ t is differentiable at t 0 (for essentially any topology in the image).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Then, if t is good for the same parameters and (M, t) is a (C a , α, β, ǫ)-admissible pair, we claim that the estimates in Lemma 2.6 hold for M and all |s| ≤ t , with constants depending only on C a . (Indeed, (39) holds for s = 0 by Proposition 2.4 since C a is larger than the constant C from that proposition, so that (34) is satisfied by the admissibility condition.) In addition, using (34) again, we may ensure, by Lemma 3.8 below, that the tower of f t coincides with that of f up to level M .…”

Section: Maps In a Neighbourhood Of A Good Map -Admissible Pairs (M T)mentioning

confidence: 92%

“…is bounded above by a constant times k max(1+2α+β/2,3β/2) |(f k−1 ) ′ (c 1 )| 1/2 |t|, where in the Misiurewicz-Thurston case the polynomial factor vanishes. Thus, in the polynomial case, by (39), we get…”

Section: 3mentioning

confidence: 96%

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Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

Baladi¹,

Benedicks

Schnellmann³

2014

Invent. math.

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We consider C 2 families t → ft of C 4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure µt of ft, as a function of t on the set of Collet-Eckmann (CE) parameters:Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters ∆, and, for each t 0 ∈ ∆, a set ∆ 0 ⊂ ∆ of polynomially recurrent parameters containing t 0 as a Lebesgue density point, and constants C ≥ 1, Γ > 4, so that, for every 1/2-Hölder function A,In addition, for all t ∈ ∆ 0 , the renormalisation period Pt of ft satisfies Pt ≤ Pt 0 , and there are uniform bounds on the rates of mixing of f Pt t for all t with Pt = Pt 0 . If ft(x) = tx(1 − x), the set ∆ contains almost all CE parameters.Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter t 0 , we find a set of CE parameters ∆ ′ M T accumulating at t 0 , a constant C ≥ 1, and a C ∞ function A 0 , so that C|t − t 0 | 1/2 ≥ A 0 dµt − A 0 dµt 0 ≥ C −1 |t − t 0 | 1/2 , ∀t ∈ ∆ ′

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Maps In a Neighbourhood Of A Good Map -Admissible Pairs (M T)mentioning

confidence: 92%

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Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

Baladi¹,

Benedicks

Schnellmann³

2014

Invent. math.

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“…It is not clear how this technique can be implemented in the presence of noise. A few recent results [2,32] used induction to achieve stochastic stability in some classes of one-dimensional maps. It would be interesting to explore the possibility of apply those techniques to metastability.…”

Section: Introductionmentioning

confidence: 99%

Escape rates formulae and metastablilty for randomly perturbed maps

Bahsoun

Vaienti²

2013

Nonlinearity

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“…Recently Shen [26] obtained stochastic stability for unimodal transformations under very weak assumptions, but does not cover the case of transformations with infinitely many critical points; and Shen together with van Strien in [27] obtained strong stochastic stability for the Manneville-Pomeaux family of intermittent maps, answering questions raised in [12].…”

Section: Introductionmentioning

confidence: 99%

Adapted random perturbations for non-uniformly expanding maps

2014

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We obtain stochastic stability of C2 non-uniformly expanding one-dimensional endomorphisms, requiring only that the first hyperbolic time map be L^{p}-integrable for p>3. We show that, under this condition (which depends only on the unperturbed dynamics), we can construct a random perturbation that preserves the original hyperbolic times of the unperturbed map and, therefore, to obtain non-uniform expansion for random orbits. This ensures that the first hyperbolic time map is uniformly integrable for all small enough noise levels, which is known to imply stochastic stability. The method enables us to obtain stochastic stability for a class of maps with infinitely many critical points. For higher dimensional endomorphisms, a similar result is obtained, but under stronger assumptions.Comment: 25 pages; 1 figure; many typos corrected. Keywords: stochastic stability, additive perturbation, non-uniform expansion, slow recurrence, adapted random perturbation, stationary measures, physical measures, first hyperbolic time map. To be published in Stochastics and Dynamic

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On stochastic stability of non-uniformly expanding interval maps

Abstract: We study the expanding properties of random perturbations of regular interval maps satisfying the summability condition of exponent one. Under very general conditions on the interval maps and perturbation types, we prove strong stochastic stability.

Cited by 16 publications

References 37 publications

Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

Escape rates formulae and metastablilty for randomly perturbed maps

Adapted random perturbations for non-uniformly expanding maps

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