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Strange attractors in higher dimensions. (English) Zbl 0784.58044
Let \(f_ \mu:M \to M\), \(\mu\in \mathbb{R}\), be a smooth one-parameter family of diffeomorphisms of an \(m\)-dimensional manifold \(M\), \(m \geq 2\), exhibiting a homoclinic tangency associated to a hyperbolic fixed (or periodic) point \(p\) of \(f_ 0\). Suppose that \(f_ 0\) is sectionally dissipative at \(p\), i.e. the product of any pair of eigenvalues of \(Df_ 0(p)\) is less than 1 in absolute value.
The purpose of this paper is to prove the following Theorem: For generic one-parameter families \((f_ \mu)\), \(\mu \in \mathbb{R}\), as above there is \(S \subset \mathbb{R}\) such that
(i) \(S \cap(-\varepsilon,\varepsilon)\) has positive Lebesgue measure for every \(\varepsilon>0\);
(ii) for all \(\mu \in S\), \(f_ \mu\) exhibits nonhyperbolic strange attractors in a (const \(| \mu |)\)-neighbourhood of the orbit of tangency.

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37D99 Dynamical systems with hyperbolic behavior
Full Text: DOI
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