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 <tex-math notation="LaTeX">μ </tex-math> , we prove that typical <tex-math notation="LaTeX">μ </tex-math> -preserving homeomorphisms have upper metric mean dimension, with respect to the Riemannian distance, equal to the dimension of the manifold. Moreover, we prove that <tex-math notation="LaTeX">μ </tex-math> is a measure of maximal metric mean dimension, with respect to the variational principle established by Velozo and Velozo.