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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fixed points of Anosov maps of certain manifolds
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by Jonathan D. Sondow PDF
Proc. Amer. Math. Soc. 61 (1976), 381-384 Request permission

Abstract:

Lemma. If $H$ is a graded exterior algebra on odd generators with augmentation ideal $J$ and $h:H \to H$ is an algebra homomorphism inducing $J/{J^2} \to J/{J^2}$ with eigenvalues $\{ {\lambda _i}\}$, then the Lefschetz number $L(h) = \Pi (1 - {\lambda _i})$. The lemma is applied to an Anosov map or diffeomorphism of a compact manifold with real cohomology $H$ to give sufficient conditions that none of the eigenvalues ${\lambda _i}$ be a root of unity and that there exist a fixed point. In particular, every Anosov diffeomorphism of a compact connected Lie group has a fixed point.
References
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 61 (1976), 381-384
  • MSC: Primary 58F15; Secondary 55C20
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0438398-7
  • MathSciNet review: 0438398