Statistical properties of unimodal maps: the quadratic family

By Artur Avila, Carlos Gustavo T. de A. Moreira. 2000

66 citations recorded by the source corpus.

We prove that almost every nonregular real quadratic map is ColletEckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step in achieving the same results for more general families of unimodal maps.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bfcc6be56d46b1034763bf86b9067170e0b177a0

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the validity of the 0–1 test for chaos (2009, background): “This family of maps is particularly well-understood [1, 12]: we can decompose the parameter interval according to [0, 4] = P ∪ C ∪ N , where N has Lebesgue measure zero and the asymptotic dynamics consists of a periodic attractor (of period q 1) for a ∈ P and a strongly chaotic attractor consisting ”
The density of prime divisors in the arithmetic dynamics of quadratic polynomials (2006, background): “(2.2) If p Disc(g ◦ fN ), then p cannot divide the discriminant of the splitting field Q(g ◦ fN ) of g ◦ fN [16, Corollary 2, p. 157], and so p is unramified in Q(g ◦ fN ).”
Regular or stochastic dynamics in real analytic families of unimodal maps (2003, result): “Those results, first obtained in the context of the quadratic family ([ AM1 ]), were generalized in [AM2], [Av] to non-trivial analytic families using results and methods of this paper, and then to generic smooth families.”