This is a sequel to my [earlier question](https://mathoverflow.net/questions/26094/a-topologically-mixing-subshift-with-multiple-measures-of-maximal-entropy), where I asked for an example of a shift space that is mixing but not intrinsically ergodic -- that is, it has multiple measures of maximal entropy (MMEs). Steve Huntsman's [answer](https://mathoverflow.net/questions/26094/a-topologically-mixing-subshift-with-multiple-measures-of-maximal-entropy/26151#26151) referred me to a paper by Haydn that gives such an example: however, in that example the two MMEs are supported on disjoint compact subsets of the shift space, and so in some sense the shift can be viewed as two intrinsically ergodic shifts that have been "glued together".


Is there an example of a transitive shift with multiple MMEs that are all fully supported? More precisely, does anybody know of a transitive shift space $X\subset \{0,1,\dots,p-1\}^\mathbb{Z}$ for which there are two distinct ergodic measures $\mu\_1, \mu\_2$ such that


1. $h\_{\mu\_1}(\sigma) = h\_{\mu\_2}(\sigma) = h\_\mathrm{top}(X,\sigma)$;
2. $\mu\_i(U)>0$ for every open set $U\subset X$ and $i=1,2$?



Transitive shifts with multiple fully supported MMEs
My recollection is that in one of the last chapters of the book by Denker-Grillenberger-Sigmund, there is a theorem which says something like: given $n$ ergodic measure-preserving transformations with entropy strictly less than $\log d$, there is a minimal subshift of the full shift on $d$ symbols which has precisely these invariant measures. This would certainly be fine if $n=2$ and two systems with identical entropy are given. Going to the Springer website, I think it's chapter 31, but unfortunately I can't get through the paywall, so I could be quite wrong...


That said, if it's not important that the topological entropy of the subshift is positive, then all that's needed is an example of a minimal subshift with zero topological entropy which is not uniquely ergodic. An example of this appears to be constructed in Section 3 of the 1984 paper "Toeplitz minimal flows which are not uniquely ergodic" by Susan Williams. 


