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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.

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  • $\begingroup$ It seems that the answer should be yes. This should rely on the fact that one can approximate subshifts by subshifts of finite type such that the entropy of the approximations approaches the entropy of the original subshift. This appears to be an old result, see for example Vaughn Climenhaga's answer to mathoverflow.net/questions/101762/… I now need to check that these entropies do exactly what I want, but I think I now know where I need to look for the answer. $\endgroup$ – Tom Kempton Oct 6 '14 at 15:06

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