Verovic, Patrick Minimal entropy problem for Finsler metrics. (Problème de l’entropie minimale pour les métriques de Finsler.) (French) Zbl 1009.53053 Ergodic Theory Dyn. Syst. 19, No. 6, 1637-1654 (1999). Let \(M\) be a compact locally symmetric space of non-compact type such that \(\text{rank}(M)>1\). Denote by \(G\) the connected component of the unit in the isometry group. The author gives explicit construction of a Finsler metric such that the total volume is equal to the Riemannian volume but that the volume growth entropy is strictly less than for a locally symmetric metric. This Finsler metric is the unique minimum for the volume growth entropy among \(G\)-invariant Finsler metrics normalized by the volume of the manifold. In the case of \(\text{rank}(M)=1\) real hyperbolic metrics are critical points for the topological entropy among all Finsler metrics normalized by the volume (by the Liouville volume in the case \(\dim(M)=2\)). Reviewer: P.Grushko (Irkutsk) Cited in 3 Documents MSC: 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics) 53C35 Differential geometry of symmetric spaces Keywords:volume; locally symmetric space; entropy; Finsler metric; volume growth entropy × Cite Format Result Cite Review PDF Full Text: DOI