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.)



A topologically mixing subshift with multiple measures of maximal entropy
Yes. See Haydn's paper [here](http://docs.google.com/viewer?a=v&q=cache%3ANF8M7-tT3w8J%3Aciteseerx.ist.psu.edu/viewdoc/download%253Fdoi%253D10.1.1.139.9001%2526rep%253Drep1%2526type%253Dpdf+Multiple+measures+of+maximal+entropy+and+equilibrium+states+for+one-dimensional+subshifts&hl=en&gl=us&pid=bl&srcid=ADGEESjeew53q334z7SrBbXQQIqOXoo1JnuKUmfHUbD9LOEa2dM8XnO6NQn_x7T8TLlF1meRY3gJxyxWMGd6G3eMYPD28D7EwfVZLYedHnwCTH_5gUX5Z3nCSiLp2JDU1jnYZW0tahWA&sig=AHIEtbRdqFxp2blJGJKe0-FjOCWRPJHjRg).


