Quotient Spaces Defined By Bijection

Question

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).

Answer 1

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:

1. 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.

2. 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.

3. 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.

4. 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.