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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 $$\lim_{n\rightarrow\infty}\ \ \frac{a_n}{\exp(n\cdot h(\varphi))}$$ 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?

4 EDIT: After Ian's example, I am adding an extra assumption that entropy is positive, that is, if we assume $h(\varphi)>0$, does the limit exist?

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 $$\lim_{n\rightarrow\infty}\ \ \frac{a_n}{\exp(n\cdot h(\varphi))}$$ 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.

3 added 30 characters in body

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. Does anyone know if limits of the form $$\lim_{n\rightarrow\infty}\ \begin{equation} \lim_{n\rightarrow\infty}\ \ \frac{a_n}{\exp(n\cdot h(\varphi))}$$ 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.

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