Typical Conservative Homeomorphisms Have Total Metric Mean Dimension

Given a compact smooth boundaryless manifold with dimension greater than one endowed with a locally positive non-atomic measure $\mu $ , we prove that typical $\mu $ -preserving homeomorphisms have upper metric mean dimension, with respect to the Riemannian distance, equal to the dimension of the manifold. Moreover, we prove that $\mu $ is a measure of maximal metric mean dimension, with respect to the variational principle established by Velozo and Velozo.