For a continuous mapping $\varphi: X \to X$ on a compact Hausdorff space $X$ we define the entropy as follows:
Given open covering $\mathcal{U}$, $\mathcal{V}$ of $X$, we call $\mathcal{V}$ a refinement of $\mathcal{U}$, if for all $V \in \mathcal{V}$ there exists $U \in \mathcal{U}$ such that $V \subseteq U$. We call
$$ \mathcal{U} \vee \mathcal{V} := \{U \cap V; \, U \in \mathcal{U}, \, V \in \mathcal{V}\}$$
the common refinement of $\mathcal{U}$ and $\mathcal{V}$.
For $n \in \mathbb{N}$ we set $\mathcal{U}_n := \bigvee_{i = 0}^{n-1} \varphi^{-i}(\mathcal{U})$, where $\varphi^{-i}(\mathcal{U}) := \{\varphi^{-i}(U); \, U \in \mathcal{U}\}$. Now denote by $N(\varphi, n, \mathcal{U})$ the minimal cardinality of a refinement of $\mathcal{U}_n$. (Notice that, since every subcover is a refinement and $X$ is compact, $N(\varphi, n, \mathcal{U})$ is always finite.) We now set
$$h_{\mathcal{U}}(\varphi) := \lim_{n \to \infty} \frac{1}{n} \log \big(N(\varphi, n, \mathcal{U})\big)$$
and call
$$ h(\varphi) = \sup \{ h_{\mathcal{U}}(\varphi) ; \, \mathcal{U} \text{ is an open covering of } X\}$$
the (topological) entropy of $\varphi$.
Notice that the constant $h_{\mathcal{U}}(\varphi)$ asymptoically satisfies
$$ N(\varphi, n ,\mathcal{U}) \sim e^{n h_{\mathcal{U}(\varphi)}}$$
for $n \to \infty$, so we can think of $h_{\mathcal{U}(\varphi)}$ as the exponential growth rate of $N(\varphi, n ,\mathcal{U})$ as $n$ grows larger.
Similarly we can define the polynomial (topological) entropy, namely by setting
$$p_{\mathcal{U}} (\varphi) := \lim_{n \to \infty} \frac{\log \big(N(\varphi, n, \mathcal{U})\big)}{\log(n)},$$
then
$$ p(\varphi) = \sup \{ p_{\mathcal{U}}(\varphi) ; \, \mathcal{U} \text{ is an open covering of } X\}$$
is called the polynomial (topological) entropy.
Similar to the classical entropy we now have asymptotically that
$$N(\varphi, n, \mathcal{U}) \sim e^{\log(n) p_{\mathcal{U}}(\varphi)} = n^{p_{\mathcal{U}}(\varphi)}.$$
This justifies the name "polynomial entropy".
Now to my question:
Is there any interesting literature that treats properties of polynomial entropy. I am particularly interested in the polynomial entropy as a topological invariant. It would also be interesting to know of any classes of topological dynamical systems for which $p(\varphi)$ is a complete topologic invariant. (An exemplary class of TDS for which the entropy is a complete topologic invariant is that of Bernoulli shifts.)
