TITLE: A follow up question related to entropy
QUESTION [6 upvotes]: 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 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?

REPLY [2 votes]: In the case of the geodesic flow acting on the unit tangent bundle of a compact negatively curved manifold, if  $a_n$ is the number of closed geodesics of length at most $n$, and $h$ the topological entropy of the geodesic flow, Margulis proved that $a_n$ is equivalent to $\frac{e^{hn}}{hn}$. 
The original article has only 2 pages, but his phd thesis was relatively recently published as a book.