Celebratio Mathematica — Birman

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Works connected to Mark Elliot Kidwell

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J. S. Birman and M. E. Kidwell: “Fixed points of pseudo-Anosov diffeomorphisms of surfaces,” Adv. in Math. 46 : 2 (November 1982), pp. 217–220. MR 679909 Zbl 0508.55001 article

This article supplements Richard Miller’s paper [1982] by establishing a result on the minimum number of fixed points in the homotopy class of a pseudo-Anosov surface transformation. We thus verify one of the many far-reaching results that William Thurston announced in a preprint [1977] dealing with the classification up to homotopy of diffeomorphisms of compact surfaces. Thurston stated his theorems without proofs, but with many hints at techniques. Fathi, Laudenbach, Poénaru, and other workers at an intensive seminar at the University of Orsay gave detailed proofs of some (but not all) of Thurston’s assertions and published their findings in a monograph [1979]. More recently, Miller [1982] and Gilman [1981] have verified Thurston’s claim that his Theorem 4 [1977, p. 81] follows from the classical investigations of Jakob Nielsen [1927, 1932]. Using Miller’s exposition of the Nielsen–Thurston theory, we establish Thurston’s Theorem 6 [1977, p. 12]:

An orientation-preserving pseudo-Anosov diffeomorphism of a closed, orientable surface has the minimum number of periodic points of every period among all maps in its homotopy class.