Poincare's influence is acknowledged by Fatou in his dynamics papers, but not
the work on 3 body problem. Fatou was inspired by another work of Poincare, namely on Kleinian groups. The analogy between iteration of rational functions (Fatou) and Kleinian groups (Poincare) also plays an important role in the modern development.
By the way, Fatou wrote the first book on Kleinian groups which is almost forgotten today, because of his extreme modesty (his name is mentioned on the title page with a small print only). The book is Appele and Goursat, Theorie des fonctions algebriques et de leurs integrales, Tome II (2-eme ed., 1930) In fact Fatou wrote
the whole second volume (Fonctions automorphes), which is of the same size as vol. I, but his contribution is only mentioned as "an editor of the second edition". The first edition was in one volume. The second volume is published a year after Fatou's death.
On the other hand, there is no doubt that Fatou was very well familiar with Poincare's work on 3 body problem, and the indirect influence of Poincare's ideas
in dynamics is felt in Fatou's work on holomorphic dynamics. (Julia was influenced by Poincare to a much lesser extent if at all).
Fatou was an astronomer by profession (he worked in Paris observatory for all his career, where his duty was to observe the stars at night). He also made an important contribution to Celestial mechanics, which is almost completely forgotten nowadays. (In particular he obtained the first rigorous results on the
"Averaging method" which is now universally credited to M. M. Bogolyubov).
But I have to add that Fatou's and Juia's main motivation in developing what we call "holomorphic dynamics" was not "dynamics", but solution of functional equations. This can be even seen from the titles of many of their papers. (Fatou's main work on dynamics is called Sur les equations fonctionnelles.)
For a comprehensive historical discussion of context of the work of Fatou and Julia I recommend the excellent book of Michele Audin "Fatou, Julia, Montel. The great prize of mathematical science of 1918 and beyond".
See also Fatou's own survey of all his work written shortly before his death.