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On a one-dimensional analogue of the Smale horseshoe. (English) Zbl 0731.58047

Author’s abstract: “We construct a transformation T: [0,1]\(\to [0,1]\) having the following properties: 1) (T,\(| \cdot |)\) is completely mixing, where \(| \cdot |\) is Lebesgue measure, 2) for every \(f\in L^ 1\) with \(\int f dx=1\) and \(\phi\in {\mathbb{C}}[0,1]\) we have \(\int \phi (T^ nx)f(x)dx\to \int \phi d\mu\), where \(\mu\) is the cylinder measure on the standard Cantor set, 3) if \(\phi\in C[0,1]\) then \(n^{- 1}\sum^{n-1}_{i=0}\phi (T^ ix)\to \int \phi d\mu\) for Lebesgue-a.e. x.”

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A99 Ergodic theory
37E99 Low-dimensional dynamical systems
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