In discussion about the question Is there a way to represent the interior of a circle with a curve?, it was mentioned that such a curve cannot be one-to-one (because $[0,1]$ is not homeomorphic to $[0,1]^2$). I'm curious about in what way the Peano curve is not one-to-one.
The construction of the Peano curve is a recursive refinement of a particular path that discretely looks one-to-one, in that it touches every coordinate point at a given scale in a bijection. In the limit there's no bijection, but at every step there is a bijection between the curve so far and the coordinates of points within $[0,1]^2$ truncated to so many binary digits.
In a surjection that is not an injection, there must be some overlap (some $x,y$ where $x\neq y$ but $f(x)=f(y)$. What I'm getting at is...where is the overlap? I'm guessing it's not just at one point - is it at all points? how much overlap? What is the nature of the overlap (for a given point on $[0,1]^2$, which points in $[0,1]$ map to it?
(for discussion's sake, use the definition of the Hilbert-Peano curve)
Edit: A small bit of clarification: given a point $(j,k)$, is their overlap, and if so, how much (what is the cardinality of the inverse image at that point)? How about just for a particular point like $(1/2, 1/2)$?