Mapping the modular flow to the first quadrant of $\Bbb R^2.$ Are the same knots produced?

Question

I'm reading: Lorenz and modular flows. I've tried to write my question to the best of my ability. As I am not an expert, I know there are mistakes here, so please correct me where I'm wrong and try provide constructive feedback. Thank you.

From my limited understanding...

We can define a dynamical system on the space of lattices using a matrix that preserves unit areas. The following rotation matrix acts on the space of lattices of unit area in $\Bbb R^2$ to produce a flow known as the modular flow, $s\in \Bbb R$:

$$A=\begin{pmatrix} e^s&0\\ 0&e^{-s} \end{pmatrix}$$

Think of lattice points in the plane flowing along families of hyperbolas of the form $y=k/x$ and $y=-k/x.$

I would like to work through a different, unorthodox setup, yet equivalent approach to analyze the modular flow. The main idea is to map these hyperbolas to the first quadrant, and then calculate the closed orbits and the knots produced.

So, instead of working with integer lattice points in the plane, i.e. $p=(x,y),$ in theory one should be able derive the modular flow for points of the form $p'=(e^x,e^y).$

Using the following special function and new rotation matrix we can map the modular flow to the first quadrant of $\Bbb R^2:$

$$\Phi_s(x)=e^{\frac{s}{\log(x)}}.$$

Notice how $\Phi$ is similar to a hyperbola because if we re-arrange the equation a bit, we see $\log(x)\log(\Phi(x))=s$ and with the substitution $u=\log(x)$ and $v=\log(\Phi(x))$ we recover the hyperbola $xy=1$ plotted in a $u-v$ coordinate system.

One of the matrices that is needed now is not $A,$ but $B:$

$$B=\begin{pmatrix} e^{-e^s}&0\\ 0&e^{-e^{-s}} \end{pmatrix}$$

All points being acted upon by the matrix $B$ flow along the families of curves $\Phi_s(x)$ where $x,\Phi \in (0,1).$

Q1: How does one derive the modular flow not for integer points in the plane, but for points of the form $(e^x,e^y)$ s.t. $x,y \in \Bbb Z?$

Q2: After reformulating the modular flow, how does one identify the closed orbits in this space?

Q3: It is known that the periodic orbits of the modular flow on the entirety of $\Bbb R^2$ produce knots in the complement of the trefoil. In fact, they link with the trefoil (the locus of degenerate lattices). So under my reformulation are the same knots produced?

Thanks so much.

Answer 1

I don't understand anything of your question.

Your link is about taking $u,v\in\Bbb{C}^*,u/v\not \in \Bbb{R}$ and defining the function $$L(s)= (e^s \Re(u) + e^{-s} \Im(u))\Bbb{Z}+(e^s \Re(v) + e^{-s} \Im(v))\Bbb{Z},\qquad s\in \Bbb{R}$$ from $\Bbb{R}$ to the set of lattices in $\Bbb{C}$,

and asking for which $u,v$ we have $L(s+T)=L(s)$ for some $T$.

It happens when $L(T)=L(0)$ which means $$\pmatrix{e^T & 0 \\ 0 & e^{-T}}\pmatrix{\Re(u) & \Re(z) \\ \Im(u) & \Im(z)}=\pmatrix{\Re(u) & \Re(z) \\ \Im(u) & \Im(z)}\pmatrix{a & b \\ c & d}$$ for some $a,b,c,d\in \Bbb{Z},ad-bc=\pm 1$.

From there we can enumerate the $u,v$ : for each $\pmatrix{a & b \\ c & d}\in GL_2(\Bbb{Z})$ if $(a+X)(d+X)-bc$ has two distinct real roots then it diagonalizes over the reals $$\pmatrix{a & b \\ c & d}=\pmatrix{\Re(u) & \Re(z) \\ \Im(u) & \Im(z)}^{-1}\pmatrix{e^T & 0 \\ 0 & e^{-T}}\pmatrix{\Re(u) & \Re(z) \\ \Im(u) & \Im(z)}$$