Flow equivalence, hyperbolic systems and a new zeta function for flows

Authors

David Fried

University of California

Article

Received:

DOI:
10.1007/BF02565860

Cite this article as:

Fried, D. Commentarii Mathematici Helvetici (1982) 57: 237. doi:10.1007/BF02565860

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Abstract

We analyze the dynamics of diffeomorphisms in terms of their suspension flows. For many Axion A diffeomorphisms we find simplest representatives in their flow equivalence class and so reduce flow equivalence to conjugacy. The zeta functions of maps in a flow equivalence class are correlated with a zeta function ζ_{H} for their suspended flow. This zeta function is defined for any flow with only finitely many closed orbits in each homology class, and is proven rational for Axiom A flows. The flow equivalence of Anosov diffeomorphisms is used to relate the spectrum of the induced map on first homology to the existence of fixed points. For Morse-Smale maps, we extend a result of Asimov on the geometric index.