Child custody and child support are they linked?

Hello, me and my son are French citizens but we live in US as permanent residents since 2 years. My son's father lives in United Kingdom and is a French citizen. I went several time to court in France and my son's father did even showed up. The judge stated that I have custody of my 8 years old son and that he can come and live in US with me. My son's father is really bad to me and he is always threatening me to take my son away from me. In France he even threaten ed my life and I filed a complain to the police. He doesn't pay child support and don't have custody. What can my son's father do against me? If he files for custody what can I do to keep my son with me? Can he file for custody even if he doesn't pay a child support?
Please help me, I really need legal advices. Thank you