Let us consider $\beta\mathbb{N}$, the Stone-Čech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ultrafilters), with the usual topology and the semigroup structure inherited (extended) from the naturals.

Also, let $\mathcal{C}$ be the (topological) Cantor set, and $\mathcal{C}^\mathcal{C}$ be the space of all functions from $\mathcal{C}$ to $\mathcal{C}$, endowed with the Tychonoff product topology (or the topology of pointwise convergence) and the semigroup structure of composition of functions.

Do you know of any characterisation of the possible topological embeddings of $\beta\mathbb{N}$ into $\mathcal{C}^\mathcal{C}$? Are there any concrete (and non-trivial) examples? Are any of these embeddings also homomorphisms?