Breaking up is always hard, regardless of who instigated it. I can understand why you have the urge to message him, and if you are on and off again kind of relationship you are bound to feel the way you do.
I would go no contact, its the best thing for you. If he broke things up, and said that it was for the best, then you need to listen to that, regardless of how things turned out in the past.
Some men hate the messaging from the ex, they feel bad that you are hurting and obliged to answer you, they may even pitty you, but some men love it for all the wrong reasons, such as ego brushing and use you for sex.
Some men will even keep tap on you, messaging you every other day, every other week, every other year etc just to see if you are still hocked on them, but in reality its for their own egocentric reasons to see if the hold they have is still present. Its torture.
Go no contact, look after you now and find a way to steer your thinking towards you rather than thinking what he may be doing.