How to understand a solenoid? 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?
 A: 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.
A: You can visualize a solenoid as part of a simple 3D dynamical system. The Wikipedia page on solenoids has excellent illustrations. If you want something even more concrete, and have some "Silly Putty" handy, you can make your own solenoid by repeating the process shown in this picture. After a few repetitions, if you're careful, you'll be holding a decent approximation to a solenoid.

