Celebratio Mathematica

R. C. Robinson and R. F. Williams: “Finite stability is not generic,” pp. 451–462 in Dynamical systems (Salvador, Brazil, 26 July–14 August 1971). Edited by M. M. Peixoto. Academic Press (New York), 1973. MR 331430 Zbl 0305.58009 incollection

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Mauricio Matos Peixoto	Related

C. Robinson and R. Williams: “Classification of expanding attractors: An example,” Topology 15 : 4 (1976), pp. 321–323. MR 415682 Zbl 0338.58013 article

There are two ways to classify attractors. The first way is to ask, when are the restrictions of two diffeomorphisms to their attractors conjugate? The second way is to ask, when are two diffeomorphisms conjugate in neighborhoods of their attractors? R. Williams has studied the first question for expanding attractors (hyperbolic structure with dimension of the attractor equal the dimension of the unstable splitting) under the assumption that the stable foliation is \( C^1 \) [1967, 1970, 1974]. In this note we give an example that shows these are different questions, i.e. we give two diffeomorphisms \( f \) and \( g \) with attractors \( \Lambda_f \) and \( \Lambda_g \) such that \[ f: \Lambda_f \to \Lambda_f \] is conjugate to \[ g: \Lambda_g \to \Lambda_g ,\] but there is not even a homeomorphism from a neighborhood of \( \Lambda_f \) to a neighborhood of \( \Lambda_g \), taking \( \Lambda_f \) to \( \Lambda_g \). (They are embedded differently.) We also exhibit a technique (for this one example) that may overcome the assumption that the stable foliation is \( C^1 \) in the work of R. Williams. We end with an appendix that proves directly that an expanding attractor is locally homeomorphic to a Cantor set cross a \( u \)-dimensional disk. Here \( u \) is the dimension of the unstable bundle.

J. Franks and B. Williams: “Anomalous Anosov flows,” pp. 158–174 in Global theory of dynamical systems (Evanston, IL, 18–22 June 1979). Edited by Z. Nitecki and C. Robinson. Lecture Notes in Mathematics 819. Springer, 1980. MR 591182 Zbl 0463.58021 incollection

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Zbigniew Henry Nitecki	Related
John Moore Franks	Related

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