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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.

##### MSC:
 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37D99 Dynamical systems with hyperbolic behavior
##### Keywords:
diffeomorphisms; strange attractors
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##### References:
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