TITLE: Transitive shifts with multiple fully supported MMEs
QUESTION [6 upvotes]: This is a sequel to my earlier question, 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 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

$h_{\mu_1}(\sigma) = h_{\mu_2}(\sigma) = h_\mathrm{top}(X,\sigma)$;
$\mu_i(U)>0$ for every open set $U\subset X$ and $i=1,2$?

REPLY [3 votes]: Poking through the DGS book mentioned in Ian's answer I came along a reference to a paper that turns out to be exactly what I wanted when I asked this question originally, so I'll post it here for the sake of closure and because it's a nice example.
The paper is by Wolfgang Krieger:  On the uniqueness of the equilibrium state, Mathematical Systems Theory 8 (2), 1974, p. 97-104.
The example is the Dyck shift, which is easiest to understand in terms of brackets.  The alphabet of the shift is a collection of $2n$ symbols that come in $n$ pairs; each pair has a left element and a right element.  So with $n=2$ we can write the four symbols as (  )  [  ].  The shift space $X$ comprises all sequences on these symbols in which the brackets are "opened and closed in the right order".  So for example, ( ) [ ] is a legal word, as is ( ( ( ) [ ] [, but ( [ [ ) is illegal because the ( bracket cannot be closed before the [ brackets are.
Sticking with $n=2$, let $B_-\subset X$ be the set of all sequences in which every left bracket has a corresponding right bracket, and $B_+$ be the set of all sequences in which every right bracket has a corresponding left bracket.  One can show that every shift-invariant measure has $\mu(B_- \cup B_+) = 1$ by partitioning the complement into a countable collection of disjoint sets indexed by the location of the first/last left/right bracket with no partner.
Define a map $\pi_+\colon B_+ \to \{0,1,2\}^\mathbb{Z}$ by sending ( to 0, [ to 1, and both ) and ] to 2.  Then $\pi_+$ is an isomorphism between the two shift spaces because every right bracket has a corresponding left bracket, and hence its identity as ) or ] is uniquely determined by the rules of the shift.  Similarly, the analogous map $\pi_- \colon B_- \to \{0,1,2\}^\mathbb{Z}$ is an isomorphism.
Because every ergodic invariant measure on $X$ is supported on either $B_-$ or $B_+$, we conclude that $h(X) = \log 3$ and that there are exactly two ergodic measures of maximal entropy $\mu_{\pm} = \nu \circ \pi_{\pm}$, where $\nu$ is the $(\frac 13, \frac 13, \frac 13)$-Bernoulli measure on the full 3-shift.  Each of these measures gives positive measure to every open set in $X$, and each is of positive entropy -- indeed, each is Bernoulli, which is part of what makes this answer so satisfying to me.
Note that for larger values of $n$ the same argument shows that $h(X) = \log(n+1)$.