A parameter ASIP for the quadratic family
Magnus Aspenberg, Viviane Baladi, Tomas Persson
TL;DR
The paper proves an ASIP in parameter space for the quadratic family near a Misiurewicz CE parameter by constructing a fat Cantor set $\\Omega_*(a_*,\\kappa_0)$ of mixing, polynomially recurrent parameters and establishing uniform decorrelation, fractional response, and Hölder regularity of the phase-space variance. It then transfers parameter sums to phase-space observables via a polynomial Benedicks–Carleson-type switching, builds block approximations and a martingale-difference scheme, and uses Skorokhod representation to couple with Brownian motion. The main result shows that, for Hölder observables with nonzero variance at $a_*$, the normalized observable sums along the critical orbit satisfy an ASIP with error exponent $\\gamma>2/5$ under normalized Lebesgue measure on small parameter intervals around $a_*$. The approach extends Schnellmann’s framework from piecewise expanding maps to unimodal dynamics by carefully controlling distortion, returns, and measure estimates on a Cantor baseline, yielding a significant step toward parameter and fractional response theories for CE quadratic maps. Potential extensions include broader CE classes, larger basins of parameters, and refinements toward full fractional response in the parameter space.
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 $\tilde \varphi_{a} := \varphi - \int \varphi \, dμ_a$, with $μ_a$ the unique acim of $T_a$, and put $(σ_a (\varphi))^2 := \int \tilde \varphi_{a}^2 \, dμ_a + 2 \sum_{i>0} \int \tilde \varphi_{a} (\tilde \varphi_{a} \circ T^i_{a}) \, dμ_a$. 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 $\varphi$ with $σ_{a_*}(\varphi)\ne 0$, there exists $ε_\varphi >0$ such that, for normalised Lebesgue measure on $Ω_*\cap [a_*-ε_\varphi, a_*+ε_\varphi]$, the functions $ξ_i(a)=\tilde \varphi_a(T_a^{i+1}(1/2))/σ_a (\varphi)$ 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.