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 $\phi$, set ${\tilde{\phi }}_a:=\phi -\int \phi d{\mu }_a$, with ${\mu }_a$ the unique acim of $T_a$, and put $({\sigma }_a(\phi ){)}^2:=\int {\tilde{\phi }}_a^{2}d{\mu }_a+2{\sum }_{i>0}\int {\tilde{\phi }}_a({\tilde{\phi }}_a\circ T_a^i)d{\mu }_a$. For any transversal mixing Misiurewicz parameter $a_{\ast }$, we find a positive measure set ${\Omega }_{\ast }$ of mixing CE parameters, containing $a_{\ast }$ as a Lebesgue density point, such that for any Hölder $\phi$ with ${\sigma }_{a_{\ast }}(\phi )\ne 0$, there exists ${\epsilon }_{\phi }>0$ such that, for normalised Lebesgue measure on ${\Omega }_{\ast }\cap [a_{\ast }-{\epsilon }_{\phi },a_{\ast }+{\epsilon }_{\phi }]$, the functions ${\xi }_i(a)={\tilde{\phi }}_a(T_a^{i+1}(1/2))/{\sigma }_a(\phi )$ satisfy an almost sure invariance principle (ASIP) for any error exponent $\gamma >2/5$. (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

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