I am taking a graduate student seminar about the Smale solenoid and it's properties. (i.e. how it is expansive, it has dense periodic points, etc). But even though it's a nice example of a weird topological space, I would still like to have some motivation on why I am showing this example. So I got to read about structural stability, and about the Smale $\Omega$-Stablity conjecture, and if it where true, it's a nice way to see Smale's solenoid structural stability since it satisfies all the properties.
My question is, is it known that the Smale Solenoid structurally stable? And if it is known, how was it proven? I mean, does it come free from a theorem or was it a hassle to prove?
Here I am going to define the Smale solenoid, and if any other definitions are necessary, just let me know and I will post them (trying to keep it as short as possible).
Given $M=\mathbb{D}\times S^1$, the solid torus, and the diffeomorfism $f:M\to M$ (diffeomorphism onto its image) given by:
$f(w,z)=(\frac{w}{8}+\frac{z}{2},z^2)$
You can see that the maximal invariant set $\Lambda$ is hyperbolic and define the solenoid map as $f\restriction_{\Lambda}$.
I apologize in advanced if this is vague but any tip would help, if you know of another motivation for presenting this example, it is also welcome (like it being a counter-example for something.)
Thanks in advanced.