Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from the complex manifold, and we can associate to it the complex entropy /real entropy as well as Julia/Fatou sets.

I presume that there is a relation between the real topological entropy and the real points of the Fatou/Julia sets. Can someone give me a reference where I can find such a relation?

For instance, is there an equivalence which says that the real topological entropy is trivial if and only if there are many real points in the Fatou set is large ? (with of course a precise notion of "many", for example with density?)