Asymptotic growth rate for primitve S-adic systems

Question

It is known that for a primitive substitution $S:\mathcal{A}\to \mathcal{A}^+$, there exists constants $c,C>0$ such that

$$ c\theta_S^n \leq \vert S^n(a)\vert \leq C \theta_S^n \quad \text{for all} \; a\in \mathcal{A} \quad \text{and} \; n\in \mathbb{N}, $$

where $\theta_S$ is the Perron-Frobenius eigenvalue of the substitution matrix associated to $S$. I was wondering whether there is similar known result for primitive $S$-adic systems. When I say primitive $S$-adic I mean a directed sequence of morphisms $(\sigma_n)_{n=0}^\infty$, $\sigma_n: \mathcal{A} \to \mathcal{A}^+$, for which there exists an $r\in \mathbb{N}$ such that the matrix associated to $\sigma_{n}\circ ... \circ \sigma_{n+r}$ has strictly positive entries.

Is there any known result stating that there exists some positive constants $\theta_{\sigma}>1$ and $c,C>0$ such that

$$ c \theta_{\sigma}^n \leq \vert \sigma_0 \circ... \sigma_n(a) \vert \leq C \theta_{\sigma}^n \quad \text{for all} \; a\in \mathcal{A} \quad \text{and} \; n\in \mathbb{N}? $$

I've been looking at the paper Beyond substitutive dynamical systems: S-adic expansions where they mention the existence of uniform frequencies for primitive S-adic systems. But they somehow don't mention these asymptotic growth rates, which I thought pop up naturally in the frequencies of substitutive systems.

Answer 1

S-adic systems are a bit of a huge topic to give a quick answer about. But the general answer to your question is that no such bound can exist in general for obvious reasons. Suppose that your directed sequence $\sigma_n$ consists of many many iterations of the substitution $\alpha: 0 \mapsto 01, 1 \mapsto 10$, then many many copies of $\beta: 0 \mapsto 011, 1 \mapsto 100$, then many more copies of the first, then many more copies of the second, etc.

Each application of $\alpha$ doubles the length, and each application of $\beta$ triples it. So if you chose the sequence correctly, there will be a subsequence where $|\sigma_0 \circ \cdots \sigma_n(a)|$ is on the order of $3^n$, and a sequence where it's on the order of $2^n$. What this really means is that there is no `single Perron eigenvalue' for general $S$-adic systems as there is in the single substitution case.

More generally, I think an answer to your question is that there are several types of properties for S-adic systems. Some, like minimality, follow immediately as in the single substitution case (requiring only the primitivity assumption you made). Some, like unique ergodicity, require assuming some sort of convergence of the matrix products so that they behave "as in the single matrix case." (Your post makes me think that you may think uniform frequencies always exist, but this is false; Berthe and Delacroix need to make assumptions for this existence.)

One way to think about this is the proof of Perron-Frobenius; it requires that the cones $M^n (\mathbb{R}^+)^d$ decrease to a line, which follows from some exponential decay. But in general, there's no reason to hope for something similar with infinitely many substitutions.

Because of the order of substitution, it's not that hard to get an analog of the Perron eigenvector and eigenvalue ON ONE SIDE (this depends on the way you define adjacency matrices; with the conventions of Berthe-Delacroix, it's a right generalized eigenvector). For instance, if you just assume that some substitution appears in your sequence infinitely many times, you'll get the right eigenvector, which gives letter frequencies.

But there's no way to get a left eigenvector in general; I think that this is related to some tough questions like the S-adic Pisot conjecture that I won't get into here. See "The S-adic Pisot conjecture on 2 letters" by Berthe-Minervino-Steiner-Thuswaldner and "Geometry, dynamics, and arithmetic of S-adic subshifts" by Berthe-Steiner-Thuswaldner.