Theorem: Let be $f$ a homeomorphism of a compact metric space $X$, then $$ h_{top}(f)=\sup_{\mu\in \mathcal{M}_{f}}~ h _\mu (f) $$

Question: The above theorem is the famous variational principle for compact spaces, I'm looking for an example to see that the hypothesis $ f $ be a homeomorphism is really necessary.

Another known theorem is

Theorem: Expansive transformations of compact metric spaces have a measure with maximal entropy.

Question: This measure is unique?

Thank you in advance.