Markov extensions and decay of correlations for certain Hénon maps. (English) Zbl 1044.37013

Benedicks, Michael; Young, Lai-Sang

Flexor, Marguerite (ed.) et al., Complex geometry and dynamical systems. Conference in honor of Adrien Douady on the occasion of his 60th birthday, Orsay, France, July 3–8, 1995. Paris: Astérisque. Astérisque 261, 13-56 (2000).

Summary: Hénon maps for which the analysis in [M. Benedicks and L. Carleson, Ann. Math. (2) 133, No. 1, 73–169 (1991; Zbl 0724.58042)] applies are considered. Sets with good hyperbolic properties and nice return structures are constructed and their return time functions are shown to have exponentially decaying tails. This sets the stage for applying the results in [L.-S. Young, Ann. Math. (2) 147, No. 3, 585–650 (1998; Zbl 0945.37009)]. Statistical properties such as exponential decay of correlations and central limit theorem are proved.
For the entire collection see [Zbl 0932.00046].

MSC:

37C40	Smooth ergodic theory, invariant measures for smooth dynamical systems
37A25	Ergodicity, mixing, rates of mixing
37C70	Attractors and repellers of smooth dynamical systems and their topological structure
37D25	Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37E20	Universality and renormalization of dynamical systems

Keywords:

Hénon map; Markov extension; decay of correlation; central limit theorem