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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Automorphisms of hyperbolic dynamical systems and $ K\sb 2$


Author: Frank Zizza
Journal: Trans. Amer. Math. Soc. 307 (1988), 773-797
MSC: Primary 58F15; Secondary 19B99, 19C99, 54H20
DOI: https://doi.org/10.1090/S0002-9947-1988-0940227-2
MathSciNet review: 940227
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Abstract: Let $ \sigma :\Sigma \to \Sigma $ be a subshift of finite type and $ \operatorname{Aut} (\sigma )$ be the group of homeomorphisms of $ \Sigma $ which commute with $ \sigma $. In [Wl], Wagoner constructs an invariant for the group $ \operatorname{Aut} (\sigma )$ using $ K$-theoretic methods. Smooth hyperbolic dynamical systems can be modeled by subshifts of finite type over the nonwandering sets. In this paper we extend Wagoner's construction to produce an invariant on the group of homeomorphisms of a smooth manifold which commute with a fixed hyperbolic diffeomorphism. We then proceed to show that this dynamical invariant can be calculated (at least $ \bmod 2$) from the homology groups of the manifold and the action of the diffeomorphism and the homeomorphisms on the homology groups.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0940227-2
Article copyright: © Copyright 1988 American Mathematical Society