Geodesic flows.

*(English)*Zbl 0930.53001
Progress in Mathematics (Boston, Mass.). 180. Boston, MA: Birkhäuser. xii, 149 p. (1999).

The main goal of the book is to present, in a self-contained way, results of the author and of Ricardo Mañé about various ways to calculate or estimate the topological entropy of the geodesic flow on a closed Riemannian manifold \(M\).

The book begins with two introductory chapters on general properties of geodesic flows including a discussion of some of its properties as a Hamiltonian system acting on the tangent bundle \(TM\) of \(M\).

The third and fourth chapters present a formula for the topological entropy of the geodesic flow in terms of asymptotic growth of the average number of geodesic arcs in \(M\) connecting two given points. This, and similar other formulas for the topological entropy are obtained as an application of a fundamental result of Y. Yomdin which is also discussed, however without proof.

The last chapter contains results, mainly due to the author, on topological conditions for \(M\) which guarantee that the topological entropy of the geodesic flow for every metric on \(M\) is positive. It is also shown that there are manifolds which satisfy these conditions, but for which the infimum of the entropies for metrics with normalized volume vanishes.

The text is accompanied by many exercises. Many of the easier details of the material are presented in this form. The writing is clear, but not very inspired.

The book begins with two introductory chapters on general properties of geodesic flows including a discussion of some of its properties as a Hamiltonian system acting on the tangent bundle \(TM\) of \(M\).

The third and fourth chapters present a formula for the topological entropy of the geodesic flow in terms of asymptotic growth of the average number of geodesic arcs in \(M\) connecting two given points. This, and similar other formulas for the topological entropy are obtained as an application of a fundamental result of Y. Yomdin which is also discussed, however without proof.

The last chapter contains results, mainly due to the author, on topological conditions for \(M\) which guarantee that the topological entropy of the geodesic flow for every metric on \(M\) is positive. It is also shown that there are manifolds which satisfy these conditions, but for which the infimum of the entropies for metrics with normalized volume vanishes.

The text is accompanied by many exercises. Many of the easier details of the material are presented in this form. The writing is clear, but not very inspired.

Reviewer: Ursula Hamenstädt (Bonn)