My feeling is that there exists an ergodic measure $\mu$ for which $G_\mu \setminus Z_\mu$ is nonempty. It is sufficient to find a uniquely ergodic subsystem which admits exceptional points for the Shannon-McMillan-Breiman theorem. I think that one can be constructed symbolically without too much difficulty by the following method.
Pick a real number $h$ lying strictly between 0 and $\log 2$, and consider a sequence $x$ in the 2-shift with the following properties:
1) For every $n \geq 1$, the sequence contains precisely $e^{nh + o(n)}$ distinct words of length $n$. (For reasons of subadditivity the $o(n)$ term is necessarily positive).
2) Every word which occurs in $x$ occurs with a well-defined frequency which is not equal to 0 or 1.
The orbit closure $X$ of such a sequence is then a uniquely ergodic subsystem of the shift with topological entropy equal to $h$. An explicit procedure for constructing such a sequence was given by Grillenberger in the 1970s (in my opinion it's not particularly hard). In particular, $X$ supports a unique invariant measure $\mu$ and $G_\mu$ includes the whole of $X$. Now, suppose that the word $x$ also satisfies the property:
3) There exists a nested sequence of subwords of $x$ such that the frequency of each of these words is less than $e^{-n(h+\varepsilon)}$ for some $\varepsilon>0$.
This implies that there is a nested sequence of cylinder sets in $X$, containing some point, such that the measures of these cylinder sets decrease at a rate faster than the "standard" local entropy $h$, and hence the point in the intersection of the cylinders belongs to $G_\mu$ but not to $Z_\mu$.
I think that there shouldn't be any problem in reconciling all three of these criteria with one another, but I will admit that I haven't attempted to write a proof of that. I think it sounds reasonable that for a larger class of measures than Gibbs measures we should have $G_\mu \subseteq Z_\mu$, but I don't have much to contribute to that end of the question...
Hi Vaughn,
It is an old result of Karl Sigmund that the space of ergodic measures of a subshift of finite type is path connected in weak* topology.
The proof is very neat and takes only a page or so.
Here is the paper:
Sigmund, Karl
"On the connectedness of ergodic systems."
Manuscripta Math. 22 (1977), no. 1, 27–32.
I don't know about generalizations. Sigmund's proof does not generalize directly.
Best Answer
Let $X=\{0,1\}^{\mathbb{N}}$ with the infinite product topology (which is metrisable). For each $n \geq 1$, define $x_n$ to be the sequence given by $x_i=0$ for $1 \leq i \leq n$, $x_i=1$ for $n+1 \leq i \leq 2n$, and $x_{2n+i}=x_i$ for all $i$. Let $T \colon X \to X$ be the shift transformation $T[(x_n)]= (x_{n+1})$. We have $T^{2n}x_n=x_n$ for every $n \geq 1$, so the measure $\mu_n$ defined by $\mu_n:=(2n)^{-1}\sum_{j=0}^{2n-1}\delta_{T^jx_n}$ is an ergodic invariant Borel probability measure for $T$. Let $\overline{0}$ denote the element of $X$ corresponding to an infinite sequence of zeroes, and similarly let $\overline{1}$ denote the infinite sequence of ones; we have $\lim_{n \to \infty} \mu_n = \frac{1}{2}(\delta_{\overline{0}}+\delta_{\overline{1}})$, and this limit is not ergodic (since the set containing only the point $\{\overline{0}\}$ has measure 1/2 but is invariant).
There is a nice paper by Parthasarathy - called, I think, "On the category of ergodic measures" - which shows that for this particular dynamical system and some of its generalisations, the set of all ergodic measures and the set of all non-ergodic measures are both weak-* dense in the set of all invariant measures, so this phenomenon can actually happen quite a lot.
(Hmm, the definition of $X$ above is supposed to have curly set brackets in it, but I can't get them to appear for some reason. Anyway, it's supposed to be the set of all one-sided infinite sequences of zeroes and ones.)