The question is "the idea * into my dreams*
In English the act of something/body entering into your dreams indicates that you are in an extreme state of Bliss/anxiety/etc.
I dreamed about her again, I just cannot get her out of my head.
I cannot stop thinking about this new installation, I even dreamed about work last night.
So it is a little different from your mother language. I think filtered is the action you suggest but it does not sound so good in this context. However, I could suggest percolate or percolated this may give the impression of something bubbling through your mind and entering your dreams. As I do not know the context you wish to use, it could be that something more harsh like Encroached on or even Violated maybe suitable. Of course, the good old fashioned "entered" could still be the best choice!
I was thinking about it so much it even entered into my dreams