Let Σp={1,…,p}ℤ be the full shift on p symbols, and let X ⊂ Σp be a subshift — that is, a closed σ-invariant subset, where $\sigma\colon \Sigma_p\to \Sigma_p$ is the left shift. Then σ is expansive, and hence there exists a measure of maximal entropy (an mme) for (X,σ).
It is well known that if X is a subshift of finite type on which σ is topologically mixing, then there is a unique mme. (See, for example, Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, 1975. In fact, Bowen proves uniqueness of equilibrium states for any Hölder continuous potential φ, but let's stick with the case φ≡0 for now.)
If X is not a subshift of finite type, then less is known. For example, if X is a β-shift, then it has a unique mme, but it is not known if this holds for subshifts that are factors of a β-shift.
Question: Does anybody know of a subshift that is topologically mixing but does not have a unique mme? (That is, a subshift that has multiple measures of maximal entropy despite being topologically mixing.)
Best Answer
Yes. See Haydn's paper here.