I am reading about Feigenbaum attractor (FA) and am getting very confused with something that is described in some books. It is written that FA is not an attractor because in its neighbourhood however small there are points of unstable periodic orbits (UPOs). But I used to think that, apart from other criteria of invariance and irreducibility, an "attractor" has an open set of points constituting basin of attraction which has non-zero measure. Now, I think FA satisfies this property as well because any open set about it is attracted to it except for a set of points (UPOs) which anyway are countable and hence of zero-measure. Am I missing something?
I do not know about the exact nature of the UPOs in question, but if they are dense, the basin of attraction cannot be an open set: In every ε-ball around a point from the basin of attraction, there is an UPO, which is hence not in the basin of attraction. This is analogous to $ℝ\backslashℚ$ not being an open set.
However, I would disagree that openness of the basin of attraction is a good criterion for an attractor.