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A parameter ASIP for the quadratic family

Aspenberg, Magnus LU ; Baladi, Viviane LU and Persson, T. O.M.A.S. LU orcid (2024) In Ergodic Theory and Dynamical Systems
Abstract

Consider the quadratic family Ta(x) = ax(1-x) for x ∈ [0, 1] and mixing Collet-Eckmann (CE) parameters a ∈ (2, 4). For bounded φ, set φ˜a := φ-∫ φ dμa, with μa the unique acim of Ta, and put (σa(φ))2 := ∫ φ˜2a dμa + 2Σ i>0 ∫ phi;˜a(φ˜a o T1a) dμa. For any mixing Misiurewicz parameter a∗, we find a positive measure set Ω∗ of mixing CE parameters, containing a∗ as a Lebesgue density point, such that for any Hölder φ with σa∗ (φ) ≠ 0, there exists ϵφ > 0 such that, for normalized Lebesgue measure on Ω∗ ∩ [a∗-ϵφ, a∗ + ϵφ], the functions ξi(a) = φ˜a(T i+1 a (1/2))/σa(φ) satisfy an almost sure invariance principle (ASIP) for any error exponent γ > 2/5. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the... (More)

Consider the quadratic family Ta(x) = ax(1-x) for x ∈ [0, 1] and mixing Collet-Eckmann (CE) parameters a ∈ (2, 4). For bounded φ, set φ˜a := φ-∫ φ dμa, with μa the unique acim of Ta, and put (σa(φ))2 := ∫ φ˜2a dμa + 2Σ i>0 ∫ phi;˜a(φ˜a o T1a) dμa. For any mixing Misiurewicz parameter a∗, we find a positive measure set Ω∗ of mixing CE parameters, containing a∗ as a Lebesgue density point, such that for any Hölder φ with σa∗ (φ) ≠ 0, there exists ϵφ > 0 such that, for normalized Lebesgue measure on Ω∗ ∩ [a∗-ϵφ, a∗ + ϵφ], the functions ξi(a) = φ˜a(T i+1 a (1/2))/σa(φ) satisfy an almost sure invariance principle (ASIP) for any error exponent γ > 2/5. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann's proof for piecewise expanding maps. We need to introduce a variant of Benedicks-Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney-Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773-844].

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author
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organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
almost sure invariance principle, central limit theorem, the quadratic family
in
Ergodic Theory and Dynamical Systems
publisher
Cambridge University Press
external identifiers
  • scopus:85205285787
ISSN
0143-3857
DOI
10.1017/etds.2024.67
language
English
LU publication?
yes
id
577f58f1-ac33-4f37-aeea-3042105d7cd7
date added to LUP
2024-12-20 09:59:26
date last changed
2025-04-04 15:17:13
@article{577f58f1-ac33-4f37-aeea-3042105d7cd7,
  abstract     = {{<p>Consider the quadratic family Ta(x) = ax(1-x) for x ∈ [0, 1] and mixing Collet-Eckmann (CE) parameters a ∈ (2, 4). For bounded φ, set φ˜a := φ-∫ φ dμa, with μa the unique acim of Ta, and put (σa(φ))2 := ∫ φ˜2a dμa + 2Σ i&gt;0 ∫ phi;˜a(φ˜a o T1a) dμa. For any mixing Misiurewicz parameter a∗, we find a positive measure set Ω∗ of mixing CE parameters, containing a∗ as a Lebesgue density point, such that for any Hölder φ with σa∗ (φ) ≠ 0, there exists ϵφ &gt; 0 such that, for normalized Lebesgue measure on Ω∗ ∩ [a∗-ϵφ, a∗ + ϵφ], the functions ξi(a) = φ˜a(T i+1 a (1/2))/σa(φ) satisfy an almost sure invariance principle (ASIP) for any error exponent γ &gt; 2/5. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann's proof for piecewise expanding maps. We need to introduce a variant of Benedicks-Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney-Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773-844].</p>}},
  author       = {{Aspenberg, Magnus and Baladi, Viviane and Persson, T. O.M.A.S.}},
  issn         = {{0143-3857}},
  keywords     = {{almost sure invariance principle; central limit theorem; the quadratic family}},
  language     = {{eng}},
  publisher    = {{Cambridge University Press}},
  series       = {{Ergodic Theory and Dynamical Systems}},
  title        = {{A parameter ASIP for the quadratic family}},
  url          = {{http://dx.doi.org/10.1017/etds.2024.67}},
  doi          = {{10.1017/etds.2024.67}},
  year         = {{2024}},
}