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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.

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The theorem is true not only for homeomorphisms but also for continuous maps that are not necessarily invertible.

For discontinuous maps $f$, I'm not sure if there's any problem beyond the fact that the definition of topological entropy is generally made under the assumption that $f$ is continuous, and so one needs to check that the definition still makes sense. I believe it does, but one should be careful that there are several definitions (spanning sets, separated sets, open covers are the three most common) which are proved to be equivalent, and it needs to be checked whether or not this proof of equivalence uses continuity. I saw a paper where if I recall correctly, it was claimed that the definitions work and the whole theorem goes through without any assumptions on continuity of $f$ -- Michaela Ciklova, ``Dynamical systems generated by functions with connected $G_\delta$ graphs'', Real Analysis Exchange 30(2), 2004/2005, pp. 617-638 -- I haven't looked closely through the argument though.

As for your second question, about uniqueness, here you need some stronger hypotheses. One easy way to get non-uniqueness is to let $(X,f)$ and $(Y,g)$ be two expansive systems with equal topological entropy, then consider their disjoint union. The mme for $(X,f)$ and the mme for $(Y,g)$ are both mmes for this new system, so you get non-uniqueness.

A more interesting question is how you get non-uniqueness in the presence of topological transitivity or other ``connectedness'' properties. This was addressed in a couple other MO questions:

A topologically mixing subshift with multiple measures of maximal entropy

Transitive shifts with multiple fully supported MMEs

There are various criteria under which you get uniqueness, and that's a broad theory that I could say more about if you want, but I think this answers the question you asked so I'll leave it here for now.

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  • $\begingroup$ @Vaughn Climenhaga Can you give me a reference to where the variational principle is stated for continuous maps (not only homeomorphisms?) – $\endgroup$
    – user11178
    Commented May 10, 2013 at 21:05
  • $\begingroup$ It's stated this way (actually in a more general form, for pressure as well) in Peter Walters, "A variational principle for the pressure of continuous transformations", American Journal of Mathematics **97**(4), 1975, pp. 937-971. I don't have Walters' book handy to see if he states it his way there as well. $\endgroup$
    – Vaughn Climenhaga
    Commented May 10, 2013 at 21:15
  • $\begingroup$ It's Theorem 8.6 in Walters. $\endgroup$
    – Ian Morris
    Commented May 10, 2013 at 21:37
  • $\begingroup$ @Vaughn Climenhaga Interesting, because in the Brin-Stuck Book and Katok Hasselblat book, they require in the statement of the theorem, that f be a homeomorphism, you know why this? I read the demonstration of this theorem, /in the book of Walters, and I seem to be the same as the other two books that I have already quoted. I'm a little confused $\endgroup$
    – user11178
    Commented May 10, 2013 at 22:38
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    $\begingroup$ I assume that the extra requirement in Brin-Stuck and Katok-Hasselblatt is purely in order to simplify certain aspects of the exposition elsewhere, because as you say, the proof of this result is the same. One simplification they may obtain (caveat - I'm writing this without either of the books at hand) is that when you define expansivity, you use a different definition for homeomorphisms than for non-invertible maps. So if you are writing an account of the whole theory, you can either do everything twice, or decide to deal strictly with homeomorphisms and not worry about optimal results. $\endgroup$
    – Vaughn Climenhaga
    Commented May 11, 2013 at 4:56
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  1. 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)

  2. 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.

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