I spent years letting my ex know that I knew that he was lying, always in a joking style because that was part of our dynamic, and wierdly I interpreted it as an honest relationship because I knew that he was lying and he knew that I knew. Nuts really
It will never be an honest relationship with this man if he is a lier, how could it be? So why would you want to maintain any form of contact? Well I know that I did because a) I was addicted to him and b) that I was used to being let down so it felt familiar.
Will he contact you again? In my experience this depends on how you presented your knowledge of his lies to him and how much he needs your attention. If he needs you and can can somehow brush off your accusations then he may come back, otherwise you have probably been devalued and discarded. Reaching out will irritate him, ignoring him will relieve him of your irritating attempts to bring him down.
After years of joking with my ex about his lies and deceipts, things changed. I was getting fed up with the situation and he didn't need me anymore. My next 'joke' was nearer the knuckle, he didn't like it, and my last communication with him was very direct, very insulting and not a joke at all. No doubt I have ceased to exist for him.
I know you don't want to believe it but it would be better for you to be in the same position. if you want to contact him let it rip, tell him the truth about himself and have no further contact.