Quotient Spaces Defined By Bijection I was working with a question in topology and came to the following statement that I can't seem to figure out:

Let $f:\mathbb R^2\rightarrow\mathbb R^2$ be a homeomorphism with no fixed points. Consider the equivalence relations $\sim$ defined as $a\sim b$ exactly when $f^n(a)=f^m(b)$ for some $n,m\in \mathbb N$. Is it true that $\mathbb R^2/\sim$ is homeomorphic to $\mathbb R\times S_1$?

My intuition here is that any such $f$ must "look like" a translation in some sense, and this is certainly true of a translation. Mainly, if I could construct a curve $\gamma:\mathbb R\rightarrow\mathbb R^2$ with $\lim_{x\rightarrow \pm\infty}\gamma(x)=\infty$ in the one point compactification of $\mathbb R^2$ and such that the image of $\gamma$ was disjoint from the image of $f\circ \gamma$, then I could  conclude, but it's not obvious to me how to do this (nor whether this is the most elegant approach to take).
 A: The answer to your question is negative (even if you assume that $f$ is orientation-preserving, otherwise a glide-reflection will yield an easy counter example). But first, some good news:


*

*By Brouwer's plane translation theorem, for each fixed-point free (I will simply say "free" below) orientation-preserving homeomorphism $h: E^2\to E^2$, and each $p\in E^2$ there exists a (proper) topological embedding $\alpha: {\mathbb R}\to E^2$ such that $\alpha(t)=p$ for some $t$, $h$ preserves the image of $\alpha$ and its action on that image is (topologically) conjugate to a translation. 


From this viewpoint, each free orientation-preserving planar homeomorphism "looks like" a translation. 


*If $f: E^2\to E^2$ is a free homeomorphism with Hausdorff quotient space, then indeed, the quotient is homeomorphic to the annulus or to the Moebius band. This is a simple application of the classification of surfaces: Every noncompact connected surface  with infinite cyclic fundamental group is homeomorphic to the annulus or Moebius band. 

*However, there are examples of orientation-preserving free planar homeomorphisms whose quotients are non-Hausdorff. The simplest examples I know are time-1 homeomorphisms of the planar Reeb flow. The  Reeb foliation of the plane is the more standard object, but there is a (smooth) flow $F_t$ on the plane whose trajectories are the leaves of the Reeb foliation. 

*Things, however, can be even worse, there are free planar homeomorphisms which cannot be embedded in any flow. 
See 
Oscillation set of a Brouwer homeomorphism, Reeb homeomorphisms by François Béguin and Frédéric Le Roux
and 
Flows of flowable Reeb homeomorphisms by Shigenori Matsumoto. 
