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Minimal entropy problem for Finsler metrics. (Problème de l’entropie minimale pour les métriques de Finsler.) (French) Zbl 1009.53053

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 \(\dim(M)=2\)).

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C35 Differential geometry of symmetric spaces
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