Your proof can be modified a bit so that it works for a general countable, amenable group $\Gamma$. In this case, we can take $B(X)$ to be the closure of $\mbox{span} \{ f \circ T_{\gamma} - f : f \in C(X), \gamma \in \Gamma \}$ and we show that $B(X)$ doesn't contain the constant functions on $X$ as follows:
I claim that every function $h \in \mbox{span} \{ f \circ T_{\gamma} - f : f \in C(X), \gamma \in \Gamma \}$ is non-negative at some point in $X$, namely $\max_{x \in X} h(x) \ge 0$. This is sometimes called the "translate property" for amenable groups, if I'm not mistaken. Anyway, after establishing that, the conclusion follows easily: it follows that one cannot uniformly approximate a negative constant (say $-7$) by a function of the above form.
The proof of the translate property I'm familiar with goes like this:
Assume for contradiction that $\max_{x \in X} h(x) = \delta < 0$ and let $\varepsilon > 0$. Let $h=\sum_{i=1}^{n}f_{i}\circ T_{\gamma_{i}}-f_{i}$. Since $\Gamma$ is amenable we can find a finite set $U \subset \Gamma$ such that $\frac{\left|\gamma_{i}U\triangle U\right|}{\left|U\right|}<\varepsilon$ for all $i=1,\dots,n$ (the Følner condition). Now consider the function $F:=\frac{1}{\left|U\right|}\sum_{u\in U}h\circ T_{u}$. Choose some point $x_0 \in X$. On the one hand, by our assumption, $F\left(x_{0}\right) = \frac{1}{\left|U\right|}\sum_{u\in U} h \left(T_{u}x_{0}\right)\le\delta<0$. On the other hand,
$\left|F\left(x_{0}\right)\right|=\left|\frac{1}{\left|U\right|}\sum_{u\in U}\sum_{i=1}^{n}f_{i}\left(T_{\gamma_{i}}T_{u}x_{0}\right)-f_{i}\left(T_{u}x_{0}\right)\right|$
By the triangle inequality, changing order of summation and using the property of U, you see that the above is at most $2n \varepsilon \max_{x\in X,1\le i\le n}\left|f_{i}\left(x\right)\right|$. So if you choose $\varepsilon$ small enough, you get a contradiction.
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
First, fix $x \in X$ and let $\mu_1 := \delta_x$ be the Dirac measure supported at $x$. Then define a sequence of probability measures $\mu_n$ such that for any $f \in C^0 (X)$, $$ \int_X f(y) \mathrm{d} \mu_n (y) = \frac{1}{n} \sum_{k=0}^{n-1} \int_X f \circ T^k (y) \mathrm{d} \mu_1 (y). $$ Apply the Banach-Alaouglu Theorem to deduce there exists a subsequence $\mu_{n_j}$ which converges in the weak-$\star$ topology. It is then very easy to prove that this limit measure is in fact T-invariant, using the formulation that $\mu$ is T-invariant if and only if $$\int_X f \circ T \mathrm{d} \mu = \int_X f \mathrm{d}\mu$$ for all continuous $f$.