A parameter ASIP for the quadratic family
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| Title: | A parameter ASIP for the quadratic family |
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| Authors: | Aspenberg, Magnus, Baladi, Viviane, Persson, T. O.M.A.S. |
| Contributors: | Lund University, Faculty of Science, Centre for Mathematical Sciences, Research groups at the Centre for Mathematical Sciences, Dynamical systems, Lunds universitet, Naturvetenskapliga fakulteten, Matematikcentrum, Forskargrupper vid Matematikcentrum, Dynamiska system, Originator |
| Source: | Ergodic Theory and Dynamical Systems. 45(3):663-703 |
| Subject Terms: | Natural Sciences, Mathematical Sciences, Mathematical Analysis, Naturvetenskap, Matematik, Matematisk analys |
| Description: | 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]. |
| Access URL: | https://doi.org/10.1017/etds.2024.67 |
| Database: | SwePub |
Nájsť tento článok vo Web of Science | 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 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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| ISSN: | 01433857 14694417 |
| DOI: | 10.1017/etds.2024.67 |