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

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Commentarii Mathematici Helvetici

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.

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Authors and Affiliations

  1. University of California, 95064, Santa Cruz, California, USA

    David Fried

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  1. David Fried
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Partially supported by MCS 76-08795.

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Fried, D. Flow equivalence, hyperbolic systems and a new zeta function for flows. Commentarii Mathematici Helvetici 57, 237–259 (1982). https://doi.org/10.1007/BF02565860

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