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M. Gromov introduced the mean dimension for a continuous map in the late1990's, which is an invariant under topological conjugacy. On the other hand,the notion of metric mean dimension for a dynamical system was introduced byLindenstrauss and Weiss in 2000 and this refines the topological entropy fordynamical systems with infinite topological entropy. In this paper we will showif $N$ is a $n$ dimensional compact riemannian manifold then, for any $a\in[0,n]$, the set consisting of continuous maps with metric mean dimension equalto $a$ is dense in $C^{0}(N)$ and for $a=n$ this set is residual. Furthermore,we prove some results related to existence and density of continuous maps onCantor sets with positive metric mean dimension and on product spaces withpositive mean dimension.

Genericity of continuous maps with positive metric mean dimension