I shall identify points on a circle with their angle in this question. For example, $0$ and $2\pi$ both correspond to the top point of the circle.

Take the function $d(\theta)=2\theta$, which is a continuous map of the circle to itself; it corresponds to going around a circle twice. $d^2$ would be going around the circle 4 times, $d^3$ to going around the circle 8 times, etc... What does it look like if we go around the circle $\infty$ times?

(Quick note: Something important to observe is that $d^{-1}(\theta)=\{\frac\theta2,\frac\theta2+\pi\}$)

To consider the question rigorously, consider the inverse limit of an infinite sequence of circles, each one connected to the previous by $d$. I could call this space the Long Circle, but that might be confused with the one point compactification of Long Line, so instead I'll call it the Eternal Round (if this space already has a name, sorry for renaming it).

How can we describe this space? Well, we can take the inverse limit on the underlying sets. Picking a point out of the Eternal Round therefore corresponds to selecting an infinite number of points from a circle, say $\theta_n$, such that $\theta_n=d(\theta_{n+1})$ (each point determines all the points previous to it). Examples of points would be $(0,0,0,\dots)$, or $(0,\pi,\frac\pi2,\frac\pi4,\dots)$. The topology on the Eternal Round is the initial topology with respect to the functions $f_i$ that maps a point of the space to $\theta_i$ on the circle.

Some properties I've been able to identify so far:

  • The space is not path connected. In fact, it has infinitely many path-connected components.
  • It can be made into an abelian topological group, using $(0,0,0,\dots)$ as identity, point-wise addition for binary operation, and point-wise negation for the inverse. This means that the space is quite symmetrical, like the circle. (This group alone is worthy of study.)
  • I think it is 1-dimensional (but I haven't been able to prove it).

My question is, what does the Eternal Round "look like"? More concretely, what is a simpler way to characterize this space? Can it be described simply in terms of previously studied spaces? Or has it been considered before?

(I'm thinking about how p-adic integers, though usually first introduced in terms of inverse limits, have a number of other easier to work with definitions and constructions, and hoping that the Eternal Round also has simpler definitions or constructions.)

  • 3
    $\begingroup$ Spaces like this have been studied. They are called solenoids. $\endgroup$ – Rob Arthan Feb 17 '17 at 1:03
  • $\begingroup$ @RobArthan ah, there we go. I didn't expect it to be embeddable in 3D space. $\endgroup$ – PyRulez Feb 17 '17 at 1:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.