I'm hoping that the question below is simple thermodynamic formalism, but I can't quite make it work. Any help would be very welcome.
Let $\Sigma:=\{0,1\}^{\mathbb N}$ and let $\Sigma^*$ be the set of finite words of $0$s and $1$s.
For each $\underline a \in \Sigma$ let $\Omega_{\underline a}$ be a countable subset of $\Sigma^*$ such that for any $\omega_1\cdots \omega_n$, $\alpha_1\cdots \alpha_m\in\Omega_{\underline a}$ the cylinders $[\omega_1\cdots\omega_n]$ and $[\alpha_1\cdots \alpha_m]$ are disjoint.
Furthermore, suppose that the sets $\Omega_{\underline a}$ are continuous in $\underline a$ in the sense that for all $n\in\mathbb N$ there exists $m\in\mathbb N$ such that if $\underline a$ and $\underline b$ have the first $m$ symbols in common then the sets of words of length $\leq n$ in $\Omega_{\underline a}$ and $\Omega_{\underline b}$ are the same.
Define a family of weighted trees $T_{\underline a}$ by letting the root of $T_{\underline a}$ be connected to vertex labelled $\omega_1\cdots\omega_n\underline a$ by an edge of length (or weight) $n$ for each $\omega_1\cdots\omega_n\in\Omega_{\underline a}$. From vertex $\omega_1\cdots\omega_n\underline a$ the tree $T_{\underline a}$ continues as tree $T_{\omega_1\cdots\omega_n\underline a}$.
Let $N_n(\underline a)$ be the number of vertices of tree $T_{\underline a}$ which are distance less than $n$ from the root (i.e. the number of vertices that we can reach from the root by following a sequence of edges whose length sums to less than $n$).
Question: Is it the case that for any shift invariant measure on $\Sigma$, we have that for $\mu$ almost every $\underline a$ the limit \begin{equation} \lim_{n\to\infty}\frac{1}{n}\log (N_n(\underline a)) \end{equation} exists?
Notes:
1) As far as I can see this question can't be turned into one about entropy of topologically mixing countable Markov shifts, as my system can't be turned into a mixing Markov shift. Nor is this directly a question about growth of number of preimages for some well defined dynamical system.
2) $\underline a\to\omega_1\cdots\omega_n\underline a$ is a contraction, which makes the continuity of the collection of sets $\Omega_{\underline a}$ particularly useful.
3) It's clear from the construction that the upper growth rate is bounded above by $\log 2$ (in particular it is finite).
4) If we were to restrict to only using edges of length less than $k$ for some $k>0$ then we could model our system by a finite Markov shift, and the corresponding growth rate exists. The only question is whether the edges of length $>k$ can cause a jump in entropy which doesn't tend to zero as $k$ grows. For mixing countable Markov shifts they can't, but we're not necessarily in that setting.
