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

**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.