# Dynamics on the torus

It is well known that a $$2-$$torus foliated with lines of irrational slopes will produce dense curves on the torus. Likewise, rational slopes will lead to a periodic orbit. However, I am not seeing the connection between these two statements (which I think are equivalent, or at least lead to equivalent results):

1. A rational slope gives rise to periodic orbits.

2. For a rational slope, the leaves of the torus are diffeomorphic to $$S^1.$$

Can someone help me see the connection between these two statements?

A rational slope of the orbit on the torus guarantees the orbit intersect and connect with itself, eventually. Thus the orbit must be diffeomorphic to $$S^1$$. Unwrap the blue orbit. See how it is diffeomorphic to $$S^1$$?

• Beautiful!!!!!!!!!!!! May 17 at 1:12
• Cool. How do you make this picture? May 17 at 7:10
• Mathematica.. May 17 at 14:47

Here is a more rigorous argument (with gaps to fill in, depending on one's disposition). Denote by $$\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$$ the (standard) $$d$$-torus and by $$\operatorname{Aff}(\mathbb{T}^d)$$ the group of affine automorphisms of it; after Endomorphisms of the Torus and Example of a continuous affine group action we have that $$\operatorname{Aff}(\mathbb{T}^d)\cong \mathbb{T}^d\rtimes \operatorname{GL}(d,\mathbb{Z})$$; that is, any affine automorphism of $$\mathbb{T}^d$$ is of the form $$p\mapsto Ap+b$$, where $$A$$ is an invertible $$d\times d$$ matrix with integer entries with determinant $$\pm 1$$ and $$b$$ is an element of $$\mathbb{T}^d$$; note that in particular any affine automorphism is a $$C^\infty$$ diffeomorphism.

For any $$\xi\in\mathbb{R}^d\setminus0$$ define a flow $$\phi^\xi_\bullet:\mathbb{R}\to \operatorname{Aff}(\mathbb{T}^d)$$ by

$$t\mapsto [p\mapsto p+t\xi],$$

where addition is interpreted as modulo $$\mathbb{Z}^d$$. It's straightforward that for any $$\xi\in\mathbb{R}^d\setminus0$$, $$\phi^\xi$$ is a locally free (= each stabilizer is discrete) $$C^\infty$$ action and its orbit foliation is precisely the $$1$$-dimensional foliation in the direction of $$\xi$$ (note that $$\phi^\xi$$ is the flow of the constant vector field $$\mathbb{T}^d\to \mathbb{R}^d, p\mapsto \xi$$; see e.g. Constant vector field on the torus $\mathbb{T}^{2n}$ is symplectic and Regarding diffeomorphism on manifolds).

In particular, when $$d=2$$ and $$\xi=(1,\sigma)$$ for $$\sigma\in\mathbb{R}_{\geq0}$$, the orbits of $$\phi^{(1,\sigma)}$$ are exactly the affine subgroups ("subheaps" or "subgrouds"; see https://ncatlab.org/nlab/show/heap) (i.e. "lines") of slope $$\sigma$$. Let us look into stabilizers of $$\phi^{(1,\sigma)}$$. If $$\phi^{(1,\sigma)}_t(p)=p$$ for $$t\neq0$$, then $$t(1,\sigma)\in\mathbb{Z}^2$$, i.e. $$t\in\mathbb{Z}\setminus0$$ and $$\sigma\in \frac{1}{t}\mathbb{Z}\leq \mathbb{Q}$$. Thus all stabilizers are trivial for irrational $$\sigma$$ and all stabilizers are isomorphic to $$\mathbb{Z}$$ for rational $$\sigma$$ (in this case if $$\sigma=\frac{p}{q}$$ is in reduced form each point will be $$q$$-periodic). In either case we have that the orbit maps $$\phi^{(1,\sigma)}_\bullet(p):\mathbb{R}\to \mathcal{O}_p, t\mapsto p+t(1,\sigma)$$ factor through $$\mathbb{R}/\mathcal{S}_p$$ and give $$C^\infty$$ injective immersions $$\mathbb{R}/\mathcal{S}_p\hookrightarrow \mathbb{T}^2$$ with images $$\mathcal{O}_p$$ (here $$\mathcal{S}_p\leq\mathbb{R}$$ is the stabilizer subgroup of $$p$$ and $$\mathcal{O}_p\subseteq \mathbb{T}^2$$ is the orbit of $$p$$). Thus the foliation consists of $$C^\infty$$ injectively immersed lines in the irrational $$\sigma$$ case (some further (standard) argument is needed here to verify that each such line is dense) and $$C^\infty$$ embedded circles in the rational $$\sigma$$ case.

In general for $$\mathbb{T}^d$$ an analogous argument can be produced. Again there is a dichotomy from the differential geometric point of view (either all leaves will be injectively immersed lines xor all leaves will be embedded circles); though from the dynamical point of view instead of a dichotomy there will be a $$d$$-chotomy. Concisely, define $$\mathcal{R}(\xi)=\{z\in\mathbb{Z}^d\,|\, z\cdot \xi = 0\}$$ to be the resonance set of $$\xi$$, where $$z\cdot\xi=\sum_{i=1}^dz_i\xi_i$$, and put $$r(\xi)=\dim_{\mathbb{Z}}(\operatorname{span}_{\mathbb{Z}}(\mathcal{R}(\xi)))$$ (one could use $$\mathbb{Q}$$ instead of $$\mathbb{Z}$$); so that $$r(\xi)$$ is the number of solutions $$z$$ of $$z\cdot\xi=0$$ linearly independent over integers (or rationals), and in particular $$r(\xi)=0$$ iff the entries of $$\xi$$ are linearly independent elements of $$\mathbb{R}$$ considered as a (n infinite dimensional) $$\mathbb{Q}$$-vector space (see e.g. What does "Consider R as an vector space over Q" mean?). Then each orbit of $$\phi^\xi$$ will be dense in an embedded $$\mathbb{T}^{d-r(\xi)}\leq \mathbb{T}^d$$ (and vice versa); in particular for $$r(\xi)=d-1$$, each orbit will be periodic. (For $$d=3$$ nice examples to think about and draw are: $$\xi^1=(1,1,1),\xi^2=(1,1,\sqrt{2}),\xi^3=(1,\sqrt{2},\sqrt{8}),\xi^4=(1,\sqrt{2},\sqrt{3})$$; observe that the leaf passing through the origin would give a good idea due to homogeneity.)