TITLE: Variational Principle for the Entropy
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: The mapping $f$ does not really need to be a homeomorphism.  The variational principle of the entropy is valid for all continuous mappings.  (See e.g., the book of Peter Walters)
A simple (boring) example of an expansive dynamical system on a compact metric space having multiple measures with maximal entropy will be the direct sum of two copies of your favorite example that has a unique measure with maximal entropy.
Another (more interesting) example is when the system has zero topological entropy but is not uniquely ergodic.  In particular, there are minimal systems that are not uniquely ergodic and have zero topological entropy.  However, one might still consider such examples pathological.
Multiplicity of measures with maximal entropy could be interpreted as some kind of "phase transition".  If measures with maximal entropy model the equilibrium states of a system (in the sense of statistical mechanics), then multiplicity of maximal entropy measures would mean the system has multiple "macroscopically distinguishable" equilibrium states.
If you allow two-dimensional dynamics (two commuting maps rather than one), Burton and Steif found examples of two-dimensional subshifts of finite type that are strongly irreducible and have more than one measures of maximal entropy.   Häggström later showed that in fact essentially any statistical mechanics model on the lattice with finite-range interactions is "equivalent" to a strongly irreducible subshift of finite type, so that the shift-invariant Gibbs measures of the former are in one-to-one correspondence with the maximal entropy measures of the latter.