  It is well known that the presence of horseshoes leads to positive entropy.
If our goal is to construct a continuous map with infinite entropy, we can
consider an infinite sequence of horseshoes, ensuring an unbounded number of
legs.
  Estimating the exact values of both the metric mean dimension and mean
Hausdorff dimension for a homeomorphism is a challenging task. We need to
establish a precise relationship between the sizes of the horseshoes and the
number of appropriated legs to control both quantities.
  Let $N$ be an $n$-dimensional compact Riemannian manifold, where $n \geq 2$,
and $\alpha \in [0, n]$. In this paper, we construct a homeomorphism $\phi: N
\rightarrow N$ with mean Hausdorff dimension equal to $\alpha$. Furthermore, we
prove that the set of homeomorphisms on $N$ with both lower and upper mean
Hausdorff dimensions equal to $\alpha$ is dense in $\text{Hom}(N)$.
Additionally, we establish that the set of homeomorphisms with upper mean
Hausdorff dimension equal to $n$ contains a residual subset of $\text{Hom}(N).$
