TITLE: How to understand a solenoid?
QUESTION [7 upvotes]: Consider the invertible extension of the circle-doubling map $T(x)=2x \pmod 1$, the new system can be represented as $X=\{(x_k)\  | \ x_{k+1}=T(x_k)\}$ (see GTM 259 M. Eindiedler & T. Ward Exercise 2.1.9) dynamically. Another representation can be described algebraically (see also GTM259 2.1.9) as the quotient of $\mathbb{R}×\mathbb{Q}_2$. A third representation may be the dual of 2-adic rational field. My question is how to image it intuitively?

REPLY [3 votes]: In my opinion, the best way to understand the solenoid intuitively is using nested intersections. 
Let $\mathbb{T}_3 \overset{def}{=} D^2 \times S^1$ where $S^1 = [0,1] \mod 1$ and $D^2 = \{z \in \mathbb{C} \mid \left|z\right| < 1\}$. Consider $f:\mathbb{T}_3 \to \mathbb{T}_3$ given by $$f(z,\theta) = (\dfrac{1}{10}z + \dfrac{1}{2}e^{i \theta},2\theta \mod 1).$$ 
Note that $f$ is well defined because $$\left| \dfrac{z}{10} + \dfrac{1}{2}e^{i \theta}\right| \leq \left|\dfrac{z}{10}\right| + \dfrac{1}{2}\left|e^{i\theta}\right| \leq \dfrac{1}{2} + \dfrac{1}{10} \leq 1.$$ Actually this implies that $f(\overline{\mathbb{T}_3}) \subset \mathrm{int} \mbox{ } \mathbb{T}_3$.
This map gives you an extension of $2x \mod 1$. Then the solenoid is the nested intersection $\Lambda = \mathop{\bigcap}\limits_{n=0}^\infty f^n(\mathbb{T}_3)$, which is the attractor of the dynamical system $(\mathbb{T}_3, f)$. Moreover $\Lambda$ is the set where $f$ is bijective. Observe that  you have a contraction in one direction and expansion in the other. Actually the map acts in the following way: Note $S^1$ spins twice inside $\mathbb{T}_3$, and the discs $D^2 \times \{\theta\}$ go to $D^2 \times \{2\theta\}$, i.e.: $$f(D^2 \times \{\theta\}) \subset D^2 \times \{2\theta\}.$$
Here you can see some pictures: http://www.matcuer.unam.mx/~aubin/vista/index4.html
This book can be illustrative as well: C. Christenson and W. Voxman, Aspects of topology. 2nd Edition, BSC Associates, Moscow, Idaho, USA, 1988, pp. 167-170. Also, it contains the inverse limit construction that @Asaf mentioned.