Let Σ_{p}={1,...,p}^{ℤ} be the full shift on p symbols, and let X ⊂ Σ_{p} 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.)