A parameter ASIP for the quadratic family
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929707843321856 |
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| author | Aspenberg, Magnus Baladi, Viviane Persson, Tomas |
| author_facet | Aspenberg, Magnus Baladi, Viviane Persson, Tomas |
| contents | Consider the quadratic family $T_a(x) = a x (1 - x)$, for $x \in [0, 1]$ and mixing Collet--Eckmann (CE) parameters $a \in (2,4)$. For bounded $φ$, set $\tilde φ_{a} := φ- \int φ\, dμ_a$, with $μ_a$ the unique acim of $T_a$, and put $(σ_a (φ))^2 := \int \tilde φ_{a}^2 \, dμ_a + 2 \sum_{i>0} \int \tilde φ_{a} (\tilde φ_{a} \circ T^i_{a}) \, dμ_a$. For any transversal 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 $σ_{a_*}(φ)\ne 0$, there exists $ε_φ>0$ such that, for normalised Lebesgue measure on $Ω_*\cap [a_*-ε_φ, a_*+ε_φ]$, the functions $ξ_i(a)=\tilde φ_a(T_a^{i+1}(1/2))/σ_a (φ)$ 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 a previous work of Baladi, Benedicks, and Schnellmann. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_12202 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A parameter ASIP for the quadratic family Aspenberg, Magnus Baladi, Viviane Persson, Tomas Dynamical Systems 37A10, 37A50, 37E05, 60F17 Consider the quadratic family $T_a(x) = a x (1 - x)$, for $x \in [0, 1]$ and mixing Collet--Eckmann (CE) parameters $a \in (2,4)$. For bounded $φ$, set $\tilde φ_{a} := φ- \int φ\, dμ_a$, with $μ_a$ the unique acim of $T_a$, and put $(σ_a (φ))^2 := \int \tilde φ_{a}^2 \, dμ_a + 2 \sum_{i>0} \int \tilde φ_{a} (\tilde φ_{a} \circ T^i_{a}) \, dμ_a$. For any transversal 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 $σ_{a_*}(φ)\ne 0$, there exists $ε_φ>0$ such that, for normalised Lebesgue measure on $Ω_*\cap [a_*-ε_φ, a_*+ε_φ]$, the functions $ξ_i(a)=\tilde φ_a(T_a^{i+1}(1/2))/σ_a (φ)$ 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 a previous work of Baladi, Benedicks, and Schnellmann. |
| title | A parameter ASIP for the quadratic family |
| topic | Dynamical Systems 37A10, 37A50, 37E05, 60F17 |
| url | https://arxiv.org/abs/2212.12202 |