References for slow entropy and netropy for Nd and Zd actions

Question

I am studying entropy. Could you give me some references for the the generalizations of entropies, both measure-theoretical and topological, for $\mathbb{Z}^d$ and $\mathbb{N}^d$ actions and for the slow-entropies associated with those actions?

Edit

My main interests are the following. This topic is related to the problem of smooth realization and to the fact that the entropy of an action can be zero even if the entropy of the generators is positive. Could you give some references about those problems?

Answer 1

Have a look for example at the works:

1. A. Katok and J. Thouvenot http://www.numdam.org/article/AIHPB_1997__33_3_323_0.pdf.

2. A. Katok, S. Katok and F. Hertz https://arxiv.org/pdf/1311.0927.pdf

This should be a good starting point.

As I mentioned, there is no local theory coming from ergodic theory. But one can reproduce all that relates to volume growth rates and certainly also the growth rate of the action on the homology.

PS: This is an answer to the original question, which was later one modified.