Let Σp={1,...,p} 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.)


2 Answers 2


Yes. See Haydn's paper here.

  • $\begingroup$ Perfect -- that's just what I was looking for! $\endgroup$ Commented May 28, 2010 at 15:48
  • $\begingroup$ Could you please update the link or give the full title or the paper? $\endgroup$
    – Leo
    Commented Oct 19, 2016 at 11:31
  • $\begingroup$ @Leo: "Multiple measures of maximal entropy and equilibrium states for one-dimensional subshifts". As Vaughn pointed out in a blog post, this appears to be an unpublished preprint of unknown date. $\endgroup$ Commented Oct 19, 2016 at 12:05

There is a book by Denker, Grillenberger and Sigmund which deals extensively with this topic: they prove a whole range of theorems which construct subshifts whose invariant measures have specified properties. In particular they construct a mixing subshift with multiple invariant measures of maximal entropy, but this is just one of many interesting constructions.

Subshifts which are not of finite type are, as a general class, quite well understood. In terms of their measurable ergodic theory, they can do pretty much anything which is not prevented by entropy. An example is the following theorem of Jewett. Suppose that $T$ is an ergodic measure-preserving transformation of a probability space $(Z,\mathcal{F},\mu)$, and the entropy of $T$ with respect to $\mu$ is finite. Let $p$ be an integer such that $\log p$ is strictly greater than the entropy of $T$ with respect to $\mu$. Then there is a subshift $X$ of $\Sigma_p$, having a unique invariant measure $\nu$, such that the measure-preserving transformation $(X,\mathcal{B},\nu,\sigma)$ is measurably isomorphic to the transformation $(Z,\mathcal{F},\mu,T)$. (Here $\mathcal{B}$ of course denotes the Borel sigma-algebra of $X$.)

  • $\begingroup$ Thanks for the reference -- I'll have to spend some time with that book in the near future... $\endgroup$ Commented May 28, 2010 at 15:49

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