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Sinai-Bowen-Ruelle measures for certain Hénon maps. (English) Zbl 0796.58025
As it was proved by the first author and L. Carleson [Ann. Math. (2) 133, No. 1, 73–169 (1991; Zbl 0724.58042)], for a positive measure set of parameters \(0<a<2\), \(b>0\) that the maps \(T_{a,b} (x,y) = (1-ax^ 2 + y,bx)\) of the real plane to itself have a topologically transitive attractor \(\Lambda_{a,b}\). The authors study statistical properties of these attractors. The main result of the paper is that for a positive measure set of parameters there is a unique Sinai-Bowen-Ruelle invariant measure (i.e., such that it has a positive Lyapunov exponent, and its conditional measures on unstable manifolds are absolutely continuous). Moreover, its support coincides with \(\Lambda_{a,b}\), and the corresponding dynamical system is Bernoulli.

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37A05 Dynamical aspects of measure-preserving transformations
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