Maybe call these Add and Remove for simplicity and consistency with ScreenMap?

You should remove the other (Add|Remove)(Position|Influence) calls, and them move the method bodies inside your new functions. This will simplify ActorMap, and ensure that people can't engineer a situation where they can move the actor without your interface firing.

This means that there will be a couple of situations where the game does more work (removing and readding actors to the position map when it doesn't need to), but the performance hit there will be negligible.