# Topological pressure for subshifts on a countable alphabet

Apologies for asking two similar questions within a week of each other, I had hoped that asking a finite alphabet version of this question would lead to enlightenment but unfortunately it didn't.

Suppose that I have a closed shift invariant set $X\subset \mathcal A^{\mathbb Z}$ where $\mathcal A$ is a countable alphabet. The shift spaces I'm interested in have three properties.

1) There is a countable list $\mathcal F$ of forbidden words such that a sequence $x\in\mathcal A^{\mathbb Z}$ is in $X$ if and only if it contains no forbidden words. (This is true because $X$ is closed).

2) There is a countable list $\mathcal T$ of allowed transitions of the form $a_{-n}\cdots a_0\to a_1$, such that $\underline x\in X$ if and only if for each $j\in\mathbb Z$ there exists $k\in\mathbb N$ such that $$x_{j-k}\cdots x_{j-1}\to x_j$$ is an allowed transition. (This is special to our situation and doesn't follow from $X$ being a subshift)

Here $n$ is not uniform, but given any choice of $a_1$ there is a natural number $N(a_1)$ such that any allowed transition to $a_1$ is of the form $a_{-n}\cdots a_0\to a_1$ where $n\leq N(a_1)$.

3) For each pair of words $a_1\cdots a_n$, $b_1\cdots b_m$ there is a path $x_1\cdots x_k$ such that

$$a_1\cdots a_nx_1\cdots x_k b_1\cdots b_m$$

is known to be allowed without knowing any further history, i.e. $a_1\cdots a_n\to x_1$ is allowed, $a_1\cdots a_n x_1\to x_2$ is allowed... This is a mixing condition.

Let $X^n\supset X$ be the countable Markov shift in which only the first $n$ words on the list $\mathcal F$ are forbidden. Denote the Gurevich pressure of this shift $P_{\phi}(\Sigma^n,\sigma)$.

Let $X_n\subset X$ be the finite Markov shift obtained by allowing only the first $n$ transitions in the list $\mathcal T$. Let $P_{\phi}(\Sigma_n,\sigma)$ denote the corresponding topological pressure.

Question: Do we have that $$\lim_{n\to\infty} P_{\phi}(\Sigma_n,\sigma)=\lim_{n\to\infty} P_{\phi}(\Sigma^n,\sigma)?$$

Motivation: I want to know that certain growth rates exist for $X$. Namely, given any history $\cdots x_{-2}x_{-1}x_0$ the weighted sums of $\phi$ over all possible futures $x_1\cdots x_n$ have a growth rate. The lower growth rate is bounded below by the pressure of $X_n$, and the upper growth rate is bounded above by the pressure of $X^n$, and so showing that the pressures are equal would show that my growth rate exists. This is related to the existence of box dimension for self-affine sets.

Notes: We can't recode our alphabet to give us a countable 1-step Markov shift, by condition 2 is basically a Markov condition, it's just that the length of the history that we need to consider in order to know whether our next symbol is allowed to be $a_n$ is not bounded independently of $a_n$. One might call this a locally Markov countable alphabet shift, although such a cumbersome name wouldn't really be helpful.

• See my answer to your other question. Look at Omri Sarig's articles about thermodynamic formalism. – Barbara Schapira Nov 16 '14 at 15:06
• Thanks Barbara, I've been trying to adapt Omri's arguments to my case, the significant difference is that I'm not dealing with a countable Markov shift, I'm dealing with a closed shift invariant subshift which may not be Markov. One can certainly define a notion of pressure on such shifts and show that it's approximated by below, but I can't show that it coincides with approximations from above, and so can't show that this notion of pressure does what I need it to. – Tom Kempton Nov 17 '14 at 10:07