So I understand what the cantor set is geometrically when referencing a line, but I am currently working on a problem that deals with the Cantor set on a circle, and for the life of me I can't visualize or conceptualize what this means.
The way my book define the cantor set was as follows: $$\bigcap_{n=0}^\infty C^n$$ Where: $$C^0=[0,1], C^1=[0,\frac{1}{3}]\cap[\frac{2}{3},1],....$$ And that makes sense to me. But what I don't get is this idea of the cantor set on a circle. So I guess I'm basically asking what each iteration of the cantor set on a circle looks like, and what it means to for the cantor set to be on something like a circle.
Due to lack of context, I can ony guess, but I suppose that the author of that text has in mind something like $\varphi(C)$, where $\varphi\colon[0,1]\longrightarrow S^1$ is the map defined by $\varphi(x)=\bigl(\cos(2\pi x),\sin(2\pi x)\bigr)$. Although $\varphi$ is not a homeomorphism, it turns out that $\varphi(C)$ and $C$ are homeomorphic.
Under this approach, that set will be again an intersection of a decreasing sequence of sets, the first three of which can be seen in the picture below.