Commentarii Mathematici Helvetici

, Volume 57, Issue 1, pp 237–259

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

• David Fried
Article

DOI: 10.1007/BF02565860

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

## 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 ζH 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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