**Definition**: Let be $f:M\to M$ a diffeomorphism of a compact manifold. We say that $A\subset M$ is an attractor when there exists a neighborhood $U\supset A$ such that $f( \overline{U})\subset int (U)$ and
$$
A=\bigcap_{n\geq 0}f^n(U)
$$
$U$ is called an basin of attraction of $f$.

**Theorem**: Let $M$ be a compact manifold and let $f:M→M$ a diffeomorphism. If
$\overline{Per f}$ has hyperbolic structure, then
can be partitioned into a finite number of compact, invariant and topologically transitive sets, called basic sets:
$$\overline{Per(f)}=⋃_{i=1}^{m}Λ_i$$

**Definition**: Le be $\Lambda_i$ a basic set of $f$, then we define $$W^s(\Lambda_i)=\{x: d(f^n(x), \Lambda_i)\to 0,~n\to\infty \}$$

**Question**: Supose that the chain recurrent set of $f$, $\mathcal{R}(f)$ has hyperbolic structure, I would like to see that if $int (W^s(\Lambda_i))\neq \varnothing$ then the basic set $\Lambda_i$ is an attractor.

nonempty interior, then Λ = M and f is Anosov" appears to be a well knownfolklore theorem. We could find no proof of it in the literature, so one is provided on p.52 – Sigrlami Jul 16 '13 at 15:09