Minimal entropy rigidity for Finsler manifolds of negative flag curvature (Q2709589)

In two papers, \textit{G. Besson}, \textit{G. Courtois} and \textit{S. Gallot} [Geom. Funct. Anal. 5, 731-799 (1995; Zbl 0851.53032); Ergodic Theory Dyn. Syst. 16, 623-649 (1996; Zbl 0887.58030)] proved the following Theorem: Let \((X,g_0)\) be a compact \(n\)-dimensional Riemannian, locally symmetric manifold of negative curvature \((n\geq 2)\) and \((Y,g)\) a compact, negatively curved, Riemannian manifold homotopy-equivalent to \((X,g_0) \). Then:NEWLINENEWLINENEWLINE(i) \(h(g_0)^n\text{Vol} (X,g_0)\leq h(g)^n\text{Vol}(Y,g)\), andNEWLINENEWLINENEWLINE(ii) equality in (i) implies that \((Y,g)\) is homotopic with \((X,g_0)\),NEWLINENEWLINENEWLINEwhere \(h(g) \) is volume growth entropy of the metric \(g\).NEWLINENEWLINENEWLINEThe authors of the present paper consider a normalized entropy functional on the space of compact, reversible Finsler manifolds \((Y,F)\) of negative flag curvature which are homotopy-equivalent to a compact Riemannian, locally symmetric manifold \((X,g_0)\) of negative curvature and show that \((X,g_0)\) is the unique minimum for this functional. In this respect one proves that the normalized entropy functional \(N(F)h(F)^n \text{Vol}(Y,F)\) verifies a relation similar to (i) and (ii) of the quoted theorem.