Indexed on: 28 Oct '20Published on: 25 Oct '20Published in: arXiv - Mathematics - Dynamical Systems
M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show some results related to existence and density of continuous maps with positive metric mean dimension and the continuity of the metric mean dimension on continuous maps defined on riemannian manifolds or Cantor sets. Furthermore, we will provide some fundamental properties of this notions.