Ok, my thoughts.
Preflop - I think this is fairly standard. I'm not going anywhere with AK but once villain minraises I'm happy to just call and see a flop, no need to race (perhaps) this early on.
Flop - A on the flop, A in my hand, the likelihood of villain having AA (a hand we have to fear in a $11 against an unknown's UTG+1 minreraise) is now statistically tiny. I check, expecting villain to bet as he's shown huge strength preflop. Villain obliges, checkraise is standard. Once he minraises back I'm not worried about just calling and letting him see the turn before getting all the chips in. Flush draws just don't play like this. Occasionally you will see a donk minraising with hands like QJs preflop, but a naked flushdraw would not 3-minbet this flop. If he calls or if he pushes, then we can add various flush draws to his range, but I give him absolutely no credit for a flush draw here. So I call. What do I put him on now? I think his range is something like AK-AQ, maybe AJ, and a high pair, say KK through TT that he's unable to get away from. I'm only worried about the possiblity of 55/88 and to a lesser extent AdKd-Jd here, which fits to an extent with the action so far. There's a possiblity of him running a bizarre bluff with two cards too, of course, but this is probably at the lower end of Harrington's 10%+ chance, but even without this I'm beating his range.
Turn - The turn pairs the board but it's a nothing card - absolutly nothing in villain's range includes a 5, except possibly 55, which is statistically very unlikely now too. I check using the same logic as on the flop - if villain is going to 3-bet me on the flop I think he's going to bet big at the turn, whether he has a huge hand, has TPQ/J kicker, has an underpair to the A on board, or has nothing and is running a bluff. I'm getting my chips in regardless, so by checking I gain value on the occasions on which villain either elects to bluff when he would have folded to a big bet, or where he bets with his marginal hand trying to push me off what he has to suspect to be an Ace of some description with his mid/high pair, again when he might have folded had I bet. On many occasions when I check the turn I'm trying to control the size of the pot - here it's the opposite - I'm trying to get all the chips in because I think I'm miles ahead of villain's pushing range given the action so far. Much like on the flop, if the worst case scenario does happen and he checks behind, I'm not overly concerned about gicing a free card, as I said earlier I only really consider AdKd to be a flush probability and whether I bet or check the turn I'm doomed on a diamond river if that is what villain has. Either way his turn push does nothing to change the range I gave him after the flop action (except that 55 is now less of a consideration).
Villain's hand wasn't in the range I gave him. Try and guess if you like.