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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

  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 $i=1,2$?
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4 Answers 4

up vote 3 down vote accepted

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.

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That paper by Williams is quite marvelous -- thanks for the reference! Going to need some time to fully digest those constructions... –  Vaughn Climenhaga Oct 30 '10 at 4:44

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 $h(X) = \log(n+1)$.

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I believe that combining these two papers, you get an answer: http://arxiv.org/abs/0906.2176 and http://arxiv.org/abs/1010.3372.

In the first one it is shown that partially hyperbolic sets with one dimensional center admit principal symbolic extension (which in particular preserve the entropy of the measures) and in the second one, an example of a topologically mixing partially hyperbolic set with two maximal entropy measures is constructed. I believe this gives an answer.

However, I do not know how to construct an explicit example.

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Thanks for the references -- those are both quite interesting. However, I'm not convinced that the paper by (Rodriguez Hertz)$^2$, Tahzibi, and Ures gives examples in which the maximal entropy measures both have full support. Certainly they get examples with multiples MMEs, and they reference the question of whether they are fully supported, but I don't see a clear answer in the exposition of their results. –  Vaughn Climenhaga Oct 30 '10 at 4:11

There is a paper by Quas and Allahbakhshi titled "Class degree and Relative Maximal Entropy," that may have an example of what you are after. The paper is available on arXiv. From their abstract:

"Given a factor code $\pi$ from a shift of finite type $X$ onto an irreducible sofic shift $Y$, and a fully supported ergodic measure $\nu$ on $Y$, we give an explicit upper bound on the number of ergodic measures on $X$ which project to $\nu$ and have maximal entropy among all measures in the fiber $\pi^{-1}\{\nu\}$. This bound is invariant under conjugacy. We relate this to an important construction for finite-to-one symbolic factor maps."

Unfortunately, they are dealing with maximal entropies among all measures in the described fiber. Of course, this does not imply maximal entropy as you requested. Perhaps it can still be of some assistance.

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Hmm... that's a different sort of problem than the one I was thinking of, but it's an interesting line of thought nonetheless. Good to know! –  Vaughn Climenhaga Oct 30 '10 at 4:48

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