Attractor

Remarks

A set that satisfies clauses 1 and 2 is often called an attracting set. The literature uses several other versions. Some replace clause 3 by the existence of a dense orbit in $A$. Some ask for a neighbourhood $U$ with $f(\overline{U}) \subseteq U$ and $A = \bigcap_{n \ge 0} f^n(U)$. Milnor's version replaces the open neighbourhood in clause 2 by a basin of positive measure.

Example: $f(x) = x/2$ on $\R$ and $A = \set{0}$. Clause 1 holds since $f(0) = 0$; clause 2 holds with $U = \R$, since $d(f^n(x), A) = \abs{x}/2^n \to 0$; clause 3 holds since $A$ has no nonempty proper subset.

Non-example: let $f$ be the time-$1$ map of the flow of $u' = u - u^3$ on $\R$, and let $A = [-1, 1]$. Every solution tends to $-1$, $0$ or $1$, so $A$ is invariant and attracts all of $\R$. But the proper subset $\set{1}$ also satisfies clauses 1 and 2, with $U = (0, \infty)$, so clause 3 fails for $A$. The attractors of this map are $\set{-1}$ and $\set{1}$.