Orders of accumulation of entropy on manifolds

For a continuous self-map $T$ of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold $M$ and any countable ordinal $\text{al}$, we construct a continuous, surjective self-map of $M$ having order of accumulation of entropy $\text{al}$. If the dimension of $M$ is at least 2, then the map can be chosen to be a homeomorphism.