Abundance of strange attractors.

*(English)*Zbl 0815.58016Consider a \(C^ \infty\) one-parameter family of surface diffeomorphisms \((f_ \varepsilon)_ \varepsilon\) and suppose that \(f_ 0\) has a homoclinic tangency associated to some periodic point \(p_ 0\). The authors show that under some open and dense assumptions there exists a positive Lebesgue measure set \(E\) of parameter values near 0, such that for \(\varepsilon \in E\), \(f_ \varepsilon\) exhibits a strange attractor, or repellor, near the orbit of tangency.

The proof is based on the observation that near the tangency the family \(f_ \varepsilon\) unfolds to a family \(\phi_ \varepsilon\) of so-called Hénon-like maps, such as small perturbations of \((x,y) \mapsto (1 - ax^ 2 + \varepsilon y,\varepsilon x)\). To such families the authors extend and generalize the famous result of Benedicks and Carleson about the existence of strange attractors in the Hénon map. This is the main part of the paper, and it presents a very careful, complete and readable account of the intricate iterative construction required to prove the strangeness of the attractor (whose existence is not a big issue). Also, some aspects of the original proof are simplified.

The proof is based on the observation that near the tangency the family \(f_ \varepsilon\) unfolds to a family \(\phi_ \varepsilon\) of so-called Hénon-like maps, such as small perturbations of \((x,y) \mapsto (1 - ax^ 2 + \varepsilon y,\varepsilon x)\). To such families the authors extend and generalize the famous result of Benedicks and Carleson about the existence of strange attractors in the Hénon map. This is the main part of the paper, and it presents a very careful, complete and readable account of the intricate iterative construction required to prove the strangeness of the attractor (whose existence is not a big issue). Also, some aspects of the original proof are simplified.

Reviewer: J.Pöschl (Zürich)

##### MSC:

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

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\textit{L. Mora} and \textit{M. Viana}, Acta Math. 171, No. 1, 1--71 (1993; Zbl 0815.58016)

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##### References:

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