Hmm, thanks, that's very clear. Maybe you can help me. I'm writing a philosophy paper about time, and I'd like to come up with a name for a pair of conditions on the time of, say, a change. So suppose an occurrence or situation E, and the time-interval AB (excuse the miserable notation, I can't do anything about it):

A) there is no stretch of time CD which is a part of AB in which E does not occur.

B) there is no stretch of time CD which is a part of AB in which E occurs.

I'd especially like to come up with a way to characterize (B), and it seemed to me that 'self-dissimilarity' might be a good way of talking about it. But upon reading your description, I think it may just not be a close enough analogy to the geometrical case.