Let $\phi$ be a homeomorphism of a topological metric space to itself. Let $A_i$ be a nested sequence of attracting sets for $\phi$. I have found out that the intersections of such sequences are called as quasi-attractors, but i don't know a lot of examples of attractors and can't find a simple example of a quasi-attractor that is not an attractor.
Definition of an attractor I am currently using:
- $\phi$-invariant compact set $K$ such that $\forall$ $U(K)$ - neighbourhood of K $\exists$ $U'(K)$ such that $\phi^n(U'(K)) \subseteq U(K)$
- $K$ has neigbourhood $N(K)$ such that $\forall x \in N(K) \: \lim_{n\to \infty} dist(\phi^n(x), K) = 0$
If I'm not mistaken first condition is still true for the limit of nested sequences, so there should be a problem with second one. Will appreciate any hint.
It seems that i had found an answer for my problem. I need a map that is contracting points to spheres that are going in limit to point. Then the sequence i need is a sequence of disks, bounded by these spheres. They are attractors, but their intersection(point) is not, because for every point $x$ in neighbourhood of it there is a sphere between centre and $x$ that is blocking the way.