## A useful example for the space of ergodic measures

Last time I discussed the following three properties that may or may not be satisfied for a map ${f}$ on a compact metric space ${X}$:

(C) The set of ergodic measures ${{\mathcal{M}_f^e}}$ is path-connected.

(D) ${{\mathcal{M}_f^e}}$ is dense in the set ${{\mathcal{M}_f}}$ of all invariant probability measures.

(H) ${{\mathcal{M}_f^e}}$ is entropy dense.

These properties are related by (H) ${\Rightarrow}$ (D) ${\Rightarrow}$ (C), and so it’s natural to ask for examples illustrating that the implications need not run the other way.

A related question that I’ve wondered about is the following: there are many examples of systems with multiple measures of maximal entropy (MMEs), such as those described by Haydn here. (This paper, Multiple measures of maximal entropy and equilibrium states for one-dimensional subshifts, appears to be an unpublished preprint of unknown date.)

However, the examples in that paper (and all other examples I knew of for multiple equilibrium states) have the property that the ergodic MMEs are supported on proper subshifts rather than being fully supported, so that in some sense one has taken two independent systems ${X}$ and ${Y}$ and glued them together into a larger shift ${Z \supset X \cup Y}$ without giving them any interaction from the point of view of the measures supported on the original systems, despite the fact that ${Z}$ is topologically mixing. Indeed, it is not hard to show that in Haydn’s example, ${{\mathcal{M}_f^e}}$ has 3 connected components: measures supported on ${X}$, measures supported on ${Y}$, and measures giving positive weight to both ${X}$ and ${Y}$.

So I wanted an example of a system with multiple measures of maximal entropy that were all fully supported. Asking on MathOverflow, I learned of various examples where the system is question is minimal and has only finitely many ergodic measures (but more than one), all with equal entropy, but this wasn’t completely satisfying to me since what I really was thinking of was a system with the sort of abundance of ergodic measures you see in the full shift.

I recently came across such an example in a 1974 paper by Krieger, which also gives an examples of a system satisfying (C) but not (D).

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.

Proposition 1 The Dyck shift has exactly two ergodic measures of maximal entropy, and each one is fully supported and Bernoulli.

Proof: 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 ${\pi_+\colon B_+ \rightarrow \{0,1,\dots,n\}^\mathbb{Z}}$ by sending the ${n}$ left brackets to the symbols ${1,\dots,n}$, and sending every right bracket to the symbol ${0}$. Then ${\pi_+}$ is an isomorphism between the two shift spaces because every right bracket has a corresponding left bracket, and hence its identity is uniquely determined by the rules of the shift. Similarly, the analogous map ${\pi_- \colon B_- \rightarrow \{0,1,\dots,n\}^\mathbb{Z}}$ is an isomorphism.

Because every ergodic invariant measure on ${X}$ is supported on either ${B_-}$ or ${B_+}$, we conclude that ${h(X) = \log (n+1)}$ and that there are exactly two ergodic measures of maximal entropy ${\mu_{\pm} = \nu \circ \pi_{\pm}}$, where ${\nu}$ is the Bernoulli measure on the full ${(n+1)}$-shift that gives equal weight to all symbols. Each of these measures gives positive measure to every open set in ${X}$. $\Box$

Proposition 2 The set of ergodic measures for the Dyck shift is path-connected, but is not dense in ${{\mathcal{M}_f}}$.

Proof: Let ${\mathcal{M}^\pm}$ denote the set of ergodic measures supported on ${B_{\pm}}$. By the isomorphism in the previous proof, each of ${\mathcal{M}^+}$ and ${\mathcal{M}^-}$ is path-connected. Moreover, because ${B_+\cap B_-}$ is a non-empty closed invariant subset of ${X}$, it supports at least one ergodic measure, hence ${\mathcal{M}^+ \cap \mathcal{M}^- \neq \emptyset}$. This shows path-connectedness of ${{\mathcal{M}_f^e}}$.

To see that ${{\mathcal{M}_f^e}}$ is not dense in ${{\mathcal{M}_f}}$, let ${\nu_1}$ be the ${\delta}$-measure supported on the fixed point ${\dots [ [ [ \dots}$, and let ${\nu_2}$ be the ${\delta}$-measure supported on the fixed point ${\dots ) ) ) \dots}$. Let ${\nu = \frac 12(\nu_1 + \nu_2)}$. Then any ergodic measure ${\mu}$ close to ${\nu}$ in the weak* topology must give weight approximately ${\frac 12}$ to each of the 1-cylinders corresponding to the symbols [ and ), and weight close to ${0}$ to the 1-cylinders corresponding to ] and (. Thus if ${x}$ is a generic point for ${\mu}$, the vast majority of symbols in ${x}$ are [ and ). However, such a sequence cannot occur in the Dyck shift, because the symbol ) cannot appear until all the preceding symbols [ have been closed with the corresponding symbol ]. This contradiction shows that ${\nu}$ cannot be weak* approximated with ergodic measures, hence (D) fails. $\Box$ 