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Embedding inverse limits of interval maps as attractors. (English) Zbl 0587.58032
Given a compact space X and $$T: X\to X$$ continuous, the inverse limit of (X,T) is the compact space $$K=\{(t_ n)|$$ $$T(t_ n)=t_{n-1}$$, $$n\geq 1\}\subset \prod^{\infty}_{0}X$$ together with the homeomorphism $$\tau ((t_ n)=(T(t_ n))$$. This dynamical system has the semidynamical system (X,T) as a factor and is the smallest such dynamical system. Also, an attractor for a homeomorphism $$f: M\to M$$ of a compact space is a closed set C such that: (1) For some open set $$U\subset M$$ with $$c| s[f(U)]\subset U$$, we have $$C=\cap^{\infty}_{n=0}f^ n(U)$$, (2) $$f| C$$ is topologically transitive. In this paper, the author shows that the inverse limit for the map $$x\to 4x(1-x)$$ of [0,1] onto itself can be embedded as an attractor into a $$C^{\infty}$$ diffeomorphism of any manifold of dimension at least 3 and into a homeomorphism of any manifold of dimension 2. The results seem to be specific to this particular example, but involve some detailed computation.
Reviewer: H.Keynes

##### MSC:
 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 28A75 Length, area, volume, other geometric measure theory
##### Keywords:
inverse system; dynamical system; attractor
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