[jrz1972]

When villain open-raises from the button, he could have almost anything since he's in an obvious steal situation. That's why we 3-bet our ATs; our hand compares favorably to his extremely broad range of raising hands.

When villain then caps, his range of hands gets narrowed down. Unless he is a maniacal bluff-raiser who can't get away from a steal attempt (and there are some out there), our hand no longer looks so good relative to his range of hands. Still, we're getting very good odds to call that preflop cap, so we do so. Folding to the cap would be a clear error.

On the flop, betting out accomplishes nothing. Most players who attempted a steal PF will automatically raise any lead-off bet in this sort of situation, so all you do is toss in two bets when you have nothing, and you get no information at all from it. Not good.

When villain checks behind, it tells us he's either playing his overcards very weakly, or he was just screwing around PF. Either way, it now becomes reasonable to bet the turn, hoping to fold him out right now.

Based on your comments in this thread, it sounds like blind defense is a problem spot for you, as it is for many players. It is easy to spew a bunch of chips by getting into pissing contests over the blinds. Don't do it. Its one thing to defend your blinds when the situation warrants, but if you go into every steal situation assuming that your opponent is on a pure bluff, these spots are going to be very -EV for you in the long run.