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 in [VV17].