Let $(X,T)$ be a one-sided topologically mixing subshift of finite type, and $g:X\to \mathbb R$ be a strictly positive, continuous function, satisfying $$\sum_{y\in T^{-1}x}g(y)=1,\quad\forall x\in X.$$ In other words $g$ is what is commonly known as a $g$-function. Define $L:C(X)\to C(X)$ by $$Lf(x)=\sum_{y\in T^{-1}x}f(y)g(y).$$ I.e. $L$ is the transfer operator of $\log g$. We call a fixed point of the adjoint $L^*$ a $g$-measure. Finally we define, for any $n\in \mathbb N$ $$\operatorname{var}_n(\log g):=\sup_{x,y\in X}\{|\log(x)-\log(y)|:x_i=y_i\forall 0\leq i\leq n\}.$$
Theorem 3.1 of [1] is the following statement:
If $g$ so defined satisfies $\sum_{n=0}^\infty \operatorname{var}_n(\log g)<\infty$, then there exists a unique $g$-measure $\mu$, and $L^n f\to \mu(f)$ uniformly for each $f\in C(X)$.
There are many ingredients in Walter's proof (which he adapted from Keane), but there is one particular portion I do not grasp:
He first shows that, for any fixed $f\in C(X)$, $\{L^nf\}_{n\in \mathbb N}$ is an equicontinuous subset of $C(X)$, which is also uniformly bounded by $\|f\|$. Thus Arzela-Ascoli guarantees the existence of a $\tilde f\in C(X)$ and subsequence $(n_k)$ such that $L^{n_k}f\to \tilde f$ in norm. Furthermore using the definition of $L$ and a simple convergence argument he shows $$\min_X(f)\leq\min_X(Lf)\leq\cdots\leq \min_X(\tilde f)\leq \max_X(\tilde f)\leq\cdots\leq \max_X(Lf)\leq \max_X (f).$$
From this we can deduce that $\min_X(\tilde f)=\min_X(L^n\tilde f)$ for any $n\in \mathbb N$. Thus for any $x\in X$ such that $L^n\tilde f(x)=\min_X(L^n\tilde f)$, we can deduce that there must exist a $x_n\in T^{-n}x$ such that $\tilde f(x_n)=\min_X(\tilde f)$. So far, so good.
However, Walters now just states that mixing guarantees that every cylinder set contains a point on which $\tilde f$ obtains its minimum. I am having trouble rigorously proving this statement. Recall that a cylinder set is of the form $C=\{(x_i)\in X:x_j=a_0,\dots, x_{n+j}=a_n\}$, i.e. it is any set where some finite sequence of coordinates of the elements are fixed. Topological mixing here means that for any non-empty, open $U, V\subset X$ there exists a $N\in\mathbb N$ such that $T^{-n}U\cap V\neq \emptyset$ for all $n\geq N$.
I believe the idea Walters has is to form some sequence of dense open sets $U_n$, such that the intersection contains only these minimization points, and use the Baire Category Theorem. However, it seems to me that we do not yet know enough to form these sets. For any $n\in\mathbb N$ we can find a finite set $\{x_0,\dots,x_n\}$ such that $x_i\in T^{-1}x_{i+1}$, and $L^i\tilde f(x_i)$ attains the minimum of $\tilde f$. What we need, for a simple proof at least, is that for any $x$ on which $\tilde f$ attains its minimum we have that $T^{-1}x$ contains points on which $\tilde f$ attains its minimum. So, in short:
How exactly does Walters conclude from mixing that every cylinder set contains a point on which $\tilde f$ attains its minimum?
[1] Walters, Peter, Ruelles’s operator theorem and g-measures, Trans. Am. Math. Soc. 214, 375-387 (1975). ZBL0331.28013.
Since $g$ is strictly positive, if $x$ is such that $L^n\tilde{f}(x) = \min_X(L^n\tilde{f})$, then all $y \in T^{-n}x$ satisfy $\tilde{f}(y) = \min_X(\tilde{f})$. Now, as noted in the paper you reference, topological mixing implies $\cup_{n=0}^\infty T^{-n}x$ is dense in $X$, so, given any cylinder set $C$, there is some $n$ with $T^{-n}x \cap C \not = \emptyset$.
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Topological mixing is an important hypothesis. For example, suppose $X$ is the set of all $(x_j)_{j=1}^\infty \in \{0,1\}^\mathbb{N}$ such that there is no $j$ with $x_j = 1, x_{j+1} = 0$. Suppose that $L^n\tilde{f}(x) = \min_X(L^n \tilde{f})$ for $x = (0,0,0,\dots)$. If $C = \{(y_1,y_2,\dots) : y_1 = 1\}$, then there is no $n$ with $T^{-n}x \cap C \not = \emptyset$.