Do convicted felons have custody rights? And if so, what is the extent of those rights?

2 years ago, my ex and I split and moved back to California from Hawaii. Our divorce was finalized in August of last year. Due to the fact that he was incarcerated at the time of the divorce, I was awarded what I asked for; Full PHYSICAL and LEGAL custody of our 8 year old daughter with NO request for Child Support. This week he was released from prison and though I would like to give him the benefit of the doubt that he will be a semi-active part of her life, I am intending to petition the court for Child Support. I am afraid that if I "ruffle feathers" that he will in turn petition the courts for Joint Custody. Though his original conviction is not of a violent nature, it is my understanding that he also had a charge against him (from before my daughter was born) of "Child Endangerment" for "kidnapping" his 3 children from his previous marriage and in my experience, he has displayed irresponsiblity & instablity in his habits. I am very concerned for my daughter's well-being.