Solenoid is hyperbolic set

Question

Let $0<\lambda<\frac{1}{2}$ and consider the map $$F(\theta,x,y):=(2\theta,\lambda x+\frac{1}{2}\cos(2\pi\theta),\lambda y+\frac{1}{2}\sin(2\pi\theta)).$$ The solenoid $\Lambda$ is defined as the maximal invariant set for $S^{1}\times D$, where $S^{1}=\mathbb{R}/\mathbb{Z}$ and $D$ is the unit disk in $\mathbb{R}^{2}$. I want to prove that $\Lambda$ is an hyperbolic set. For the definition of a hyperbolic set, see Brin & Stuck, page 108. The solenoid is also described in this book in more detail (Chapter 1).

It is easy to calculate the derivative: $$DF(\theta,x,y)=\begin{pmatrix} 2&0&0\\ -\pi\sin(2\pi\theta)&\lambda&0\\ \pi\cos(2\pi\theta)&0&\lambda \end{pmatrix}.$$ How do I find the unstable and stable subspaces? I think that I have to use proposition 5.4.3 (page 115) in the book linked above, i.e. find a cone field.

Also, why isnt it enough to find the eigenspaces? Why do we need such techinical propositions?

Answer 1

In general the eigenspaces are not sufficient because they need not be invariant under the dynamics.

There is thus no escape from the formulas giving the stable and unstable spaces as intersections of images and preimages of cones under appropriate derivatives.

However, in your example (which is a simple prototype of more general solenoids), the space of vectors of the form $(0,*,*)$ is invariant under the derivatives. This implies that they generate the stable spaces (for which thus you don't need the intersection formulas).

On the other hand, it is very difficult to describe explicitly the unstable spaces.