论文标题
二次家族的参数ASIP
A parameter ASIP for the quadratic family
论文作者
论文摘要
考虑二次家族$ t_a（x）= a x（1-x）$，对于$ x \ in [0，1] $和混合collet-eckmann（ce）参数$ a \ in（2,4）$。对于有限的$φ$，设置$ \tildeφ_{a}：=φ-\intφ\，dμ_a$，带有$μ_a$ $ t_a $的唯一ACIM，然后put $（σ_a（φ）） \tildeφ_{a}（\tildeφ_{a} \ circ t^i_ {a}）\，dμ_a$。 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在$ω_*\ cap [a _* - ε_φ，a _*+ε_φ] $上，函数$ξ_i（a）= \tildeφ_a（t_a^{i+1}（1/2）（1/2））/σ_a（φ）$几乎可以确保$ 5 $ user（ASIP）$ 2/σ_a（φ）$。 （尤其是，伯克霍夫总计满足了这一ASIP。）我们的论点沿着施内尔曼（Schnellmann）的划分扩展地图的证明。我们需要引入Benedicks-Carleson参数排除的一种变体，并利用Baladi，Benedicks和Schnellmann的先前工作中相关性的分数响应和统一的指数衰减。
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.
