Celebratio Mathematica

R. F. Williams: “The ‘\( \mathrm{DA} \)’ maps of Smale and structural stability,” pp. 329–334 in Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 14. American Mathematical Society (Providence, RI), 1970. MR 264705 Zbl 0213.50303 incollection

The purpose of this paper is to describe in detail the “derived from Anosov” maps of the title (due to S. Smale [1967, p. 789]) and to show how these examples allow one to lower by 1 the dimension in the following theorem of Smale [1966]:

There is an open set \( U \) in the space of \( C^r \) vector fields (\( r > 0 \)) on a 4-dimensional manifold such that no \( X \in U \) is structurally stable.

The resulting theorem is definitive in that M. Peixoto has shown [1962] that structurally stable systems on compact two manifolds are dense. The present construction is based on a one-dimensional attractor (generalized solenoid [Williams 1967]) whereas Smale’s is based on a two-dimensional attractor (a torus). Otherwise this construction is just like that of Smale’s. The paper of Peixoto and Pugh [1968] contains another variation of Smale’s construction for noncompact 2-manifolds and has a good discussion as to how all of these examples work.

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R. F. Williams: “Classification of one dimensional attractors,” pp. 341–361 in Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 14. American Mathematical Society (Providence, RI), 1970. MR 266227 Zbl 0213.50401 incollection

This is a sequel to a paper [1967] in which one dimensional attractors are characterized. Some familiarity with [1967] is assumed. Attractor is meant in the sense of S. Smale; for a general introduction to this subject, see the important paper [Smale 1967]. As this is written, the axioms A and B of Smale [1967, pp. 777–778] have been shown to be nongeneric by Abraham and Smale [1970]. Thus the formulation of Smale (sometimes called Anosov–Smale systems or diffeomorphisms) will have to be varied again. But for technical reasons, the example of Abraham and Smale does not directly affect work done on attractors. Moreover, the analysis carried out here and in [1967] does concern an open set of diffeomorphisms, and thus will be pertinent to any formulation of this theory.

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R. F. Williams: “Zeta function in global analysis,” pp. 335–339 in Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 14. American Mathematical Society (Providence, RI), 1970. MR 266252 Zbl 0213.50402 incollection

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