I do not know what is a Fatou/Julia set of a diffeomorphism of a real manifold.

The usual definitions of Fatou/Julia sets are applied to rational map of a 1-dimensional complex manifold, namely the Riemann sphere.

The topological entropy of such map is always equal to log of the degree, and this is not related to its Fatou or Julia set.

I do not know what is a Fatou/Julia set of a diffeomorphism of a real manifold.

The usual definitions of Fatou/Julia sets are applied to rational map of a 1-dimensional complex manifold, namely the Riemann sphere.

The topological entropy of such map is always equal to log of the degree, and this is not related to its Fatou or Julia set.

EDIT Aug 23: Similar situation persists in higher dimension (in holomorphic setting) MR2026895 Gromov, Mikhail, On the entropy of holomorphic maps. Enseign. Math. (2) 49 (2003), no. 3-4, 217–235. The topological entropy of a surjective holomorphic self-map of the complex projective space is equal to the logarithm of the topological degree.

Again, this has nothing to do with Fatou/Julia sets.

I do not know what is a Fatou/Julia set of a diffeomorphism of a real manifold.

The usual definitions of Fatou/Julia sets are applied to rational map of a 1-dimensional complex manifold, namely the Riemann sphere.

The topological degree of such map is always equal to log of the degree, and this is not related to its Fatou or Julia set.

I do not know what is a Fatou/Julia set of a diffeomorphism of a real manifold.

The usual definitions of Fatou/Julia sets are applied to rational map of a 1-dimensional complex manifold, namely the Riemann sphere.

The topological entropy of such map is always equal to log of the degree, and this is not related to its Fatou or Julia set.