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Proceedings of the American Mathematical Society

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

 
 

 

Fixed points of Anosov maps of certain manifolds


Author: Jonathan D. Sondow
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
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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.


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Keywords: Anosov diffeomorphism, periodic point, exterior algebra, Lie group, Lefschetz number, eigenvalue, root of unity
Article copyright: © Copyright 1976 American Mathematical Society