For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim\_{n\rightarrow\infty}\frac{1}{n}\log a\_n,$$ where in each case $a\_n$ represents some appropriate quantity (see, for example, [this answer](https://mathoverflow.net/questions/69218/if-you-were-to-axiomatize-the-notion-of-entropy/69231#69231) to one of my previous questions.) Let $h(\varphi)$ denote a typical entropy that is defined by a limit as above and **after** Ian's example, assume that $h(\varphi)>0$. Does anyone know if limits of the form 
\begin{equation}
\lim\_{n\rightarrow\infty}\ \ \frac{a\_n}{\exp(n\cdot h(\varphi))}
\end{equation} have been studied anywhere? I will appreciate any possible information about such limits. For example, is there a known case where the limit exists? If so, what is the limit called? etc.


**EDIT**: As pointed out later by Ian, even if we assume $h(\varphi)>0$ this limit may not exist. I was curios to know if there were instances where the limit is known to exist. Or even better, can one characterize self-maps $\varphi$ for which the limit exists? 



A follow up question related to entropy
The limit exists for the first two examples that come to mind, namely topological entropy on the full shift and on certain simple Markov shifts.


If $X \subset \Sigma\_d^+ = \{1,2, \dots, d\}^{\mathbb{N}}$ and $\sigma$ is the shift map, then for the topological entropy the quantity $a\_n$ denotes the number of words of length $n$ that appear in some sequence $x\in X$. If $X$ is the full shift, then $a\_n = d^n$, the entropy is $h = \log d$, and we quickly see that $a\_n / e^{nh} = 1$ for all $n$.


Slightly more interesting is when you have a Markov shift, say $X\subset \Sigma\_2^+$ determined by the transition matrix $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$.


In this case it's not hard to show that the sequence $a\_n$ is actually the Fibonacci sequence, and thus writing $\phi = \frac {1+\sqrt 5}2$ and $\psi = \frac { 1-\sqrt 5}2$, we have
$$
a\_n = \frac 1{\sqrt 5} (\phi^{n+2} - \psi^{n+2}).
$$
Since $|\psi|<1$, this shows that the limit of $a\_n / e^{nh}$ exists.


My guess is that a similar argument works for other Markov shifts and shows that the limit exists in those cases, based on obtaining a recurrence relation for $a\_n$ and then an exact formula using standard tools for solving such sequences.


All that said, it's not immediately clear what the significance of the limit is, and I don't know of any name for it. For other interesting shifts, such as sofic shifts or shifts with specification, I'd be surprised if the limit always exists.


What **is** certainly quite important is to have conditions under which the ratio $a\_n / e^{nh}$ is bounded away from $0$ and $\infty$. Such estimates are a significant part of arguments on the uniqueness of a measure of maximal entropy (and more generally uniqueness of equilibrium states), in particular the proof that such a measure satisfies a Gibbs property. For example, see Bowen's 1975 paper "[Some systems with unique equilibrium states](https://doi.org/10.1007/BF01762666 "Math. Syst. Theory 8(1974), 193-202 (1975). zbMATH review at https://zbmath.org/0299.54031")". (Dan Thompson and I also struggled with this not too long ago in Section 5.1 of [this paper](https://arxiv.org/abs/1011.2780 "Climenhaga, Vaughn; Thompson, Daniel J. Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors. Isr. J. Math. 192, Part B, 785-817 (2012), doi:10.1007/s11856-012-0052-x. zbMATH review at https://zbmath.org/1279.37011").)


It turns out that in the general setting, one of those bounds is immediate -- the sequence $a\_n$ is submultiplicative, and so it's not hard to show that $a\_n \geq e^{nh}$ for all $n$, whatever other properties the shift space has. Getting an upper bound on $a\_n / e^{nh}$ is harder and requires some sort of specification property.


