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 \(n\geqslant 2\), 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 \(\text{Hom}(N).\)

众所周知，马蹄铁的存在会导致正熵。如果我们的目标是构造一个具有无限熵的连续地图，我们可以考虑无限的马蹄铁序列，确保无限数量的腿。

估计同胚的度量平均维数和平均豪斯多夫维数的精确值是一项具有挑战性的任务。我们需要在马蹄铁的尺寸和相应的腿的数量之间建立精确的关系来控制这两个数量。

令\(N\)为\(n\)维紧致黎曼流形，其中\(n\geqslant 2\)和\(\alpha\in[0,n]\)。在本文中，我们构造了一个同态\(\phi:N\rightarrow N\)，其平均豪斯多夫维数等于\(\alpha\)。此外，我们证明了在\(N\)上的同态集，其下均 Hausdorff 维数和上均 Hausdorff 维数都等于\(\alpha\)的同胚集在\(\text{Hom}(N)\)中是稠密的。此外，我们确定上均 Hausdorff 维数等于\(n\)的同态集合包含\(\text{Hom}(N).\)的残差子集。