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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

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
28A75 Length, area, volume, other geometric measure theory
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