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I am trying to show that the orbit of the function $T:\mathbb{T}^2\to\mathbb{T}^2$ defined as $T(x,y)=(x+\alpha,y+\beta)$ has interesting properties dependent on your choice of $\alpha$ and $\beta$, notably that when $\alpha /\beta$ is irrational the orbit is dense in $\mathbb{T}^2$. I am going to try to prove it without the use of any argument except that the orbit is dense in the one dimensional case. My issue I’m stuck on currently is setting up $\mathbb{T}^2:=\mathbb{R}^2/\mathbb{Z}^2$ as a metric space. I am not very confident in quotient topologies so my current attempt looks like this:

Let $d_2$ be the usual metric on $\mathbb{R}$, then define $d_{\mathbb{T}^2}(X,Y)=\min\limits_{\underline{x}\in X,\underline{y}\in Y}\{d_2(\underline{x},\underline{y})\}$ for all $X,Y\in\mathbb{T}^2$.

I believe this is a metric, although I am struggling to prove the triangle inequality. Am I headed in the right direction, or is there a much easier way of doing this?

Alp Uzman
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Noah Gilbertson
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  • Do you not know that the torus can be embedded in $\mathbb{R}^3$ (or, somewhat easier, $\mathbb{R}^4$)? Or that it is homeomorhic to $S^1 \times S^1$? – Ben Steffan Feb 07 '25 at 16:31
  • I think it is most appropriate with the skills I currently have and the nature of the problem, to view it as the topological group I have shown here. It’s basically the cross product of two real numbers mod 1. – Noah Gilbertson Feb 07 '25 at 16:46
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    Sure, view it as a topological group. Show that, as topological groups, it is isomorphic to $S^1 \times S^1$, and conclude from there. This is the "much easier way of doing this" you were asking about. – Ben Steffan Feb 07 '25 at 16:49

1 Answers1

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Indeed you are on the right track, and my humble opinion is that for dynamical problems this perspective is important, while perhaps not being the slickest approach as others commented. This perspective is also important from a metric geometry perspective as I understand.

Thus $\mathbb{T}^d = \mathbb{R}^d/{\mathbb{Z}^d}$ is the $d$-torus as a factor topological (or Lie) group. Its elements are cosets of the integer lattice:

$$ \mathbb{T}^d = \{x+\mathbb{Z}^d | x\in\mathbb{R}^d\},\quad\quad x+\mathbb{Z}^d = \{x+n\in\mathbb{R}^d | n\in\mathbb{Z}^d\}. $$

Your attempt in defining the metric on the factor group is accurate; put

$$ d(x+\mathbb{Z}^d,y+\mathbb{Z}^d) = \min\{|(x+n)-(y+m)| | n,m\in\mathbb{R}^d\}. $$

For the triangular inequality, let $x,y,z\in \mathbb{R}^d$, $n,m,k\in\mathbb{Z}^d$. Then by the triangular inequality applied to the Euclidean metric gives

$$ |(x+n)-(y+m)| \leq |(x+n)-(z+k)| + |(z+k)-(y+m)|. $$

Taking the minimums first over $k$ then over $n$ and $m$ provides the triangular inequality for $d$.

Note that the metric on the torus you defined coincides with the so-called Pompeiu-Hausdorff metric between the cosets; for $(X,d)$ a metric space and $A,B\subseteq X$ two nonempty closed subsets one defines their Pompeiu-Hausdorff distance between them by

$$ d_{\mathcal{H}(X)}(A,B) = \max \{d(A\leftarrow B), d(B\leftarrow A)\}, $$

where

$$ d(A\leftarrow B) = \sup_{a\in A}\inf_{b\in B} d(a,b). $$

(See Proof for the existence of a compact attractor for more on this metric. Further, last semester I taught a class where we made intensive use of the PH metric in the context of dynamics and fractals; see https://youtu.be/xLx4fWO_ktU?feature=shared&t=2432)

The PH metric is very useful in dynamics, especially in situations where the quotient space is of dubious structure; see e.g. Gogolev's paper "Partially hyperbolic diffeomorphisms with compact center foliations" for a very nice application.

Yet another construction that coincides with your definition is the quotient semi-metric on $X$, where $(X,d)$ is a metric space and $R$ is an equivalence relation on $R$. For $x,y\in X$, put

$$ d_R(x,y) = \inf \left\{\left.\sum_{i=1}^k d(a_i,b_i)\right| x=a_1, b_k = y, b_i\sim_R a_{i+1}\right\}. $$

Thus to compute the $d_R$ distance between $x$ and $y$, one tries to go from $x$ to $y$ in finitely many steps, except that at each step one can readjust the point in its $R$-equivalence class with no additional cost. One can then identify any two points in $X$ whose $d_R$-distance is $0$ to obtain the factor $X/R$. (See Burago, Burago & Ivanov's A Course in Metric Geometry for details and definitions.)

This idea again is very important in dynamics; for instance the notion of accessibility in the theory of partial hyperbolicity is based on this semimetric. Alternatively, in foliation theory $d_R$ is closely related to the idea of leaf metrics (here $R$ is a foliation).

Finally since you are ultimately interested in rotations on tori, here is a discussion regarding the higher dimensional case: Dynamics on the torus

Alp Uzman
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