Sinai-Bowen-Ruelle measures for certain Hénon maps

0. Introduction. We study maps T a;b : R 2 ! R 2 deened by T a;b (x; y) = (1 ax 2 + y; bx); 0 < a < 2; b > 0: In BC2] it was proved that for a positive measure set of parameters (a; b), T a;b has a topologically transitive attractor = a;b on which there is some hyperbolic behavior. The aim of this paper is to study the statistical properties of these attractors. Using the machinery developed in BC2] we prove the following Theorem. There is a set R 2 with Leb(() > 0 such that for all (a; b) 2 , T = T a;b admits a unique SBR measure. Moreover, supp () = and (T;) is Bernoulli. A T-invariant Borel probability measure is called a Sinai-Bowen-Ruelle (SBR) measure if there is a positive Lyaponov exponent-a.e. and the conditional measures of on unstable manifolds are absolutely continuous with respect to the Riemannian measure induced on these manifolds. (A more precise deenition is given in Section 3.4.1.) This notion is due to Sinai S1], S2]. See also LS]. The signiicance of these measures is evident in the following corollary. Corollary. For (a; b) 2 , T = T a;b has the following property: Let U be a neighborhood of. Then there is a set ~ U U with positive Lebesgue measure such that for all continuous functions ' : U ! R, 1 n n1 X i=0 ' T i (x) ! Z ' dd for every x 2 ~ U. This corollary follows from our theorem and general nonuniform hyperbolic theory (see PS,Thm.1] or Section 4.1.2). The property expressed in this corollary is often assumed to be true in physics; it is also taken for granted in numerical experiments. In the case of Axiom A attractors, mathematical justiication of this property is provided by the theory of Sinai, Bowen and Ruelle (see e.g. B], Ru1] and S2]). It has been conjectured that many other attractors admit SBR measures, but as far as we know, the H enon family and similar examples (see next paragraph) are the only nonuniformly hyperbolic attractors for which SBR measures have been constructed. Mora and Viana MV] proved recently that in homoclinic bifurcations of surface dif-feomorphisms, very small attractors with high periods appear for a positive measure set