Well, I may have a different response than others. I absolutely didn't want my ex around my son. He was dangerous physically, verbally, emotionally, mentally, and even spiritually! I did want him to have even one second of visitation. My ex was not involved during my pregnancy, and after I had Noah, he saw him about 3 times before he was 4 mos. old. Then he got married, and then got stationed in Washington state (the northwestern most tip of Washington…he couldn't have been farther away unless he was in Alaska! I thank God for that everyday!). Before he left, though, I sought the counsel of a lawyer. In Florida, they are extremely father friendly, whether the guy deserves it or not. They've been known to actually give full custody to convicted child molesters.
So the lawyer wasn't very hopeful in me getting full custody. On top of the state's general father friendly (or 50/50 they call it) policy, our area just got a new "head judge" who was so father friendly that it was disgusting what he was doing to many children.
I wasn't sure if I should send pictures and letters to my ex…if that would help (he frequently played the game that if I sought his attention, he'd ignore me, and if I backed away, he'd seek my attention). So at first I thought maybe I should try and get him to come over, so that's what I did when he was on leave a couple times. I called and asked if he'd be coming everyday, told him when to come so he could help with Noah's "fussy time" (newborn who cried from 6-8pm every night, lol), and I even, reluctantly, agreed to let my son be in my ex's WEDDING PICTURES with his new wife!! I was so sick to my stomach that day. It was the most repulsive thing I've seen, really. So anyway, my ex did exactly what I'd hoped. When I called, he would get kind of panicked when I asked if he'd be coming everyday, etc. As I said, he only saw him about 3 times, and that was all before he was 4 mos. old.
I got a couple calls before he was 2 yrs old (no money, no cards, no presents, nothing), but he never ever asked about my son. he did manage to ask for a picture of ME one time, though!!! What an idiot.
So anyway, if it were me in that situation, and I felt it was seriously not healthy for my child, I'd play the game whichever way I thought would get my child out of their life.