A parameter ASIP for the quadratic family

Abstract

Consider the quadratic family $T_a(x)=ax(1-x)$, for $x\in [0,1]$ and mixing Collet--Eckmann (CE) parameters $a\in (2,4)$. For bounded $\ddot{I}†$, set $\tilde{\ddot{I}}{†}_a:=\ddot{I}†-\int \ddot{I}†d\hat{I}{¼}_a$, with $\hat{I}{¼}_a$ the unique acim of $T_a$, and put $(\ddot{I}{ƒ}_a(\ddot{I}†){)}^2:=\int \tilde{\ddot{I}}{†}_a^{2}d\hat{I}{¼}_a+2{\sum }_{i>0}\int \tilde{\ddot{I}}{†}_a(\tilde{\ddot{I}}{†}_a\circ T_a^i)d\hat{I}{¼}_a$. For any transversal mixing Misiurewicz parameter $a_{\ast }$, we find a positive measure set $\hat{I}{\text{c◯}}_{\ast }$ of mixing CE parameters, containing $a_{\ast }$ as a Lebesgue density point, such that for any Hölder $\ddot{I}†$ with $\ddot{I}{ƒ}_{a_{\ast }}(\ddot{I}†)\ne 0$, there exists $\hat{I}{\mu }_{\ddot{I}}†>0$ such that, for normalised Lebesgue measure on $\hat{I}{\text{c◯}}_{\ast }\cap [a_{\ast }-\hat{I}{\mu }_{\ddot{I}}†,a_{\ast }+\hat{I}{\mu }_{\ddot{I}}†]$, the functions $\hat{I}{¾}_i(a)=\tilde{\ddot{I}}{†}_a(T_a^{i+1}(1/2))/\ddot{I}{ƒ}_a(\ddot{I}†)$ satisfy an almost sure invariance principle (ASIP) for any error exponent ${\hat{I}}^3>2/5$. (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.

Version v2 is the electronic copy of the version to appear in ETDS