Rigidité topologique sous l'hypothèse \flqq entropie majorée\frqq\ et applications (Q951967)

Summary: We study some families of compact length spaces whose entropy is bounded from above. We prove that these families are complete w.r.t. the Gromov-Hausdorff distance and we give an explicit constant \(\varepsilon _{0} > 0\) such that, on balls of radius \(\epsilon _{0}\) with respect to the Gromov-Hausdorff distance, the fundamental group is constant, the universal covers are close for the equivariant Gromov-Hausdorff distance, the length spectrum is continuous and the entropy is Lipschitz. If we consider now some subsets of manifolds, we show moreover that the volume is semi-continuous from below and that the integral of the Ricci curvature is bounded from below.