Statistical properties of unimodal maps: The quadratic family. (English) Zbl 1078.37029

Avila, Artur; Moreira, Carlos Gustavo

Ann. Math. (2) 161, No. 2, 831-881 (2005).

Here, the dynamics of a typical nonregular quadratic map \(f_a=a-x^2\), where \(-1/4\leq a\leq 2\) is a parameter, is studied from the statistical point of view. The work describes the asymptotic behavior of its critical orbit. The main results are stated in the following two theorems.
Theorem A: Almost every nonregular real quadratic map satisfies the Collet-Eckmann condition \[ \liminf_{n\to\infty}\dfrac{\ln (| Df^n(f(0))| )}{n}>0. \] Theorem B: Almost every nonregular real quadratic map has polynomial recurrence of the critical orbit with exponent 1, \[ \limsup_{n\to\infty}\dfrac{-\ln (| f^n(0)| )}{\ln (n) }=1. \] In other words, the set of \(n\) such that \(| f^n(0)| <n^{-\gamma}\) is finite if \(\gamma >1\) and infinite if \(\gamma <1\).
Theorem B appears to be the first result on polynomial estimates on the recurrence of the critical orbit valid for a positive measure set of nonhyperbolic parameters and answers affirmatively a long standing conjecture of Ya. Sinai. Theorems A and B show that typical quadratic maps have good ergodic properties, as exponential decay of correlations and stochastic stability in the strong sense. The results obtained are an important step in achieving the same results for more general families of unimodal maps.
The paper contains a detailed description of up-to-date results on quadratic families of unimodal maps that enriches the paper exposition.

Reviewer: Eugene Ershov (St. Petersburg)

MSC:

37E05	Dynamical systems involving maps of the interval
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A25	Ergodicity, mixing, rates of mixing

Keywords:

nonregular real quadratic maps; unimodal interval maps; Collet-Eckmann condition; stochastic behavior