Very elegant! exactly was I was looking for, Thank you! Just one more question: is $X^{G}$ a CW complex in this case?

Oh, sorry I did not see you comment! Unfortunately I don't have a concrete and clear motivation. I found this question interesting in it self (at least for me). I would love to hear the answer to your second question too!

Unfortunately, my knowledge about dynamics is limited! I will be grateful if you can give more details! thank you nay way!

In your first question there was two questions. 1) Morita invariance imply dependence only on derived categories? 2)Cannot different algebras have equivalent derived categories without having equivalent categories of modules

I'm not sure I understand your remarks. 1) Here, It is about derived Morita equivalences. 2) The point is that we are considering differential graded algebras and not only algebras.

You are right Peter! I need to find a good formulation, it is like there is 19 geometries, and one of them is parametrized by natural numbers.

Wonderful, Thank you for the reference! Just for the future reader: the total classification is summarized in pages 129-131. The answer is: There is 19 geometries in dimension 4!

It will be nice if you can give more details!!