A parameter ASIP for the quadratic family

TL;DR

This paper establishes an almost sure invariance principle for Birkhoff sums in the quadratic family at certain parameters, extending probabilistic limit theorems to a broad class of dynamical systems.

Contribution

It introduces a new parameter exclusion method and leverages fractional response and decay of correlations to prove ASIP for quadratic maps at CE parameters.

Findings

ASIP holds for Birkhoff sums at many CE parameters

Almost sure invariance principle with error exponent > 2/5

Extension of probabilistic limit theorems to quadratic maps

Abstract

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 $\varphi$, set $\tilde \varphi_{a} := \varphi - \int \varphi \, d\mu_a$, with $\mu_a$ the unique acim of $T_a$, and put $(\sigma_a (\varphi))^2 := \int \tilde \varphi_{a}^2 \, d\mu_a + 2 \sum_{i>0} \int \tilde \varphi_{a} (\tilde \varphi_{a} \circ T^i_{a}) \, d\mu_a$. For any transversal mixing Misiurewicz parameter $a_*$, we find a positive measure set $\Omega_*$ of mixing CE parameters, containing $a_*$ as a Lebesgue density point, such that for any H\"older $\varphi$ with $\sigma_{a_*}(\varphi)\ne 0$, there exists $\epsilon_\varphi >0$ such that, for normalised Lebesgue measure on $\Omega_*\cap [a_*-\epsilon_\varphi, a_*+\epsilon_\varphi]$, the functions $\xi_i(a)=\tilde \varphi_a(T_a^{i+1}(1/2))/\sigma_a (\varphi)$ 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.