×

zbMATH — the first resource for mathematics

Topological entropy of geodesic flows on simply connected manifolds, and related topics. (English. Russian original) Zbl 0904.58047
Izv. Math. 61, No. 3, 517-535 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 3, 57-74 (1997).
The author relates the topological entropy \(h(g)\) of a smooth metric \(g\) on a simply connected closed manifold \(M\) to the topological complexity of \(M\). For this he defines a metric invariant \(D^h(M,g)\) of \(g\) which measures the stretch of Lipschitz maps from the standard simplex into \(M\) which formally combine to special singular chains representing a nontrivial homology class. He then uses an estimate of the rank of the homotopy group \(\pi_{k+ 1} (M)\) be the \(k\)-th Betti number of the space of curves with fixed endpoints of length at most \(kD^h(M,g) -\text{diam} (M,g)+ \varepsilon\) to derive the inequality \(h(g)\geq D^h (M,g)^{-1} \lim \sup_{k\to\infty} {1\over k} \log (rk\pi_k(M))\).
Some applications are given. The paper also contains some discussion on estimates for the systole of \(g\). Major parts of the paper are expository.

MSC:
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
54C70 Entropy in general topology
57R65 Surgery and handlebodies
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
PDF BibTeX XML Cite
Full Text: DOI