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- W2062185873 abstract "There have been many results in recent years showing that the topological entropy of a diffeomorphism [5], [16], [23], or a continuous map [11], [12], [17], is at least the logarithm of certain eigenvalues of the map it induces in real homology. These relationships, conjectured by Shub [21, p. 37], are part of an effort to find connections between the dynamics of a discrete system and the topology of the underlying manifold [20], [22]. There do not seem to have been any results of this type for flows because the time one map of any flow is isotopic to the identity, making the natural generalization of Shub's conjecture vacuous. In this paper we present two results linking the topological entropy h(q) of the geodesic flow qS on a compact Riemannian manifold M with the exponential growth rate X of volume in the universal cover M: (1) h(o) ? X and (2) h(q) = X if M has non-positive curvature. The technique is to look for separated and spanning sets of geodesics using the Hopf-Rinow theorem for (1) and the way geodesics spread out in the case of non-positive curvature for (2). Our first result may be regarded as a strengthening of Section 4 of [8] where Dinaburg showed that h(q) > 0 if w1(M) has exponential growth while the second simplifies [13] where Margulis began by construcing the measure of maximal entropy [14]. I should like to thank the staff at IMPA Rio de Janeiro, where this work was done, for their hospitality during the year 1977-78 and Manfredo do Carmo, Ricardo Mahe6 and Floris Takens for helpful remarks." @default.
- W2062185873 created "2016-06-24" @default.
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- W2062185873 date "1979-11-01" @default.
- W2062185873 modified "2023-10-16" @default.
- W2062185873 title "Topological Entropy for Geodesic Flows" @default.
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- W2062185873 doi "https://doi.org/10.2307/1971239" @default.
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