Consider J. Milnor's paper: On the concept of attractor.
There he writes: "A less restrictive definition" [than some of the previous ones he had considered] of the concept of an attract is "Liapunov stability".
He continues: "However, I believe that not every Liapunov stable set should be called an attractor. Here is an example. Consider a diffeomorphism or a flow on the plane which reduces to the identity map on a collection of concentric circles converging to the origin, but which pushes points slightly away from the origin otherwise. Then the origin is Liapunov stable [...] although it does not attract any other point."
I propose here the follow informal and tongue-in-cheek meta-definition: I will call definition of an attractor noteworthy if a famous mathematician proposed it. This paper of Milnor contains several noteworthy definitions (but, it seems, no exhaustive comparison among them).
My question is, is there an good definition of attractor, that is not yet a Liapunov stable set, yet encompasses all other noteworthy definitions?
The reason I am asking this is, that is is extremely cumbersome to write up lecture notes, where I develop the theory for one type of attractor, only to discover that actually a different definition would have been better suited - so I then need to rewrite everything.
If I would have one general definition -even if it is unnatural- I could write up everything for that and then only specialize at certain points to more specific definitions of attractors (but using Liapunov stable sets does seem too much).
