Integrable geodesic flows on the suspensions of toric automorphisms (Q2783048)

The authors continue their investigation of the geodesic flow on a suspension of an automorphism of a torus, begun in an earlier paper [Invent. Math. 140, 639-650 (2000; Zbl 0985.37027)]. In this note they show that for automorphisms \(A\in SL(n,\mathbb{Z})\) with only real eigenvalues the measure entropy of the geodesic flow with respect to any smooth invariant measure vanishes, and the topological entropy is bounded from below by the sum NEWLINE\[NEWLINE\sum_{ \lambda_j \text{ eigenvalue, }|\lambda_j |>1}\log|\lambda_j |.NEWLINE\]NEWLINE They also work out in detail the example when \(n=2\) and \(A=\left( \begin{smallmatrix} 2 & 1 \\ 1 & 1\end{smallmatrix} \right)\). In this case the topological entropy of the flow can be explicitly computed. Moreover there are two four-dimensional submanifolds \(N^u\), \(N^v\) whose complement is foliated by invariant tori and such that \(N^u \cap N^v\) is of dimension 3 and invariant, with restricted flow of full topological entropy.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00006].