Flow equivalence, hyperbolic systems and a new zeta function for flows
- 68 Downloads
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.
Unable to display preview. Download preview PDF.
- [B2]Bowen, Rufus,Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Springer-Verlag LNM 470, 1975.Google Scholar
- [B3]—.Topological entropy and Axiom A, Global Analysis (Proc. Symp. Pure Math XIV, Berkeley, Calif. 1968) AMS, Providence 1970.Google Scholar
- [DGS]Denker, M., C. Grillenberger andK. Sigmund,Ergodic theory on compact spaces, Springer-Verlag LNM 527, 1976.Google Scholar
- [FLP]Fathi, A., F. Laudenbach andV. Poenaru (Eds.)Travaux de Thurston sur les Surfaces-Seminaire Orsay. Asterisques66–67 (1979).Google Scholar
- [Fr]Franks, John.Anosov diffeomorphisms, Global Analysis (op. cit.).Google Scholar
- [F1]Fried, David.Cross-sections to flows, Ph.D. thesis, U. Calif. Berkeley, 1976.Google Scholar
- [F2]Fried, David,Geometry of cross-sections to flows, to appear in Topology.Google Scholar
- [G]Gallovotti, G. Zeta functions and basic sets, IHES preprint.Google Scholar
- [T]Thurston, W. P. On the geometry and dynamics of diffeomorphisms of surfaces, preprint.Google Scholar
- [W]Williams, R. F. The structure of Lorenz attractors, Publ. Math. IHES50 (1980).Google Scholar