I've been exploring the composition of essentially the Dedekind $\eta$-function with parabolic Möbius transformations,
$$C_L(z,t)=\left(\frac{z}{-tz+1}\right)^{\frac{1}{2}}\eta\left(\frac{z}{-tz+1}\right)\circeq\exp(\frac{i2\pi t}{24})\: z^{\frac{1}{2}}\:\eta(z)=C_R(z,t),$$
where the symbol $\circeq$ is used to signify that equality holds only for integer $t$, and I came across the interesting parametric curves below for $z=-2+.3i$ and $-12 \leq t \leq 12$ :
$xL(t)=Real[C_L(z,t)]\:\:\:$ and $\:\:\:yL(t)=Imag[C_L(z,t)]$ and analogously for $C_R(z,t)$
Inverting
$C_L(z,t)$ is annihilated by $\frac{\partial }{\partial t}-z^2\frac{\partial }{\partial z}$, while $C_R(z,t)$ is not, even at integer $t$.
Letting $z \mapsto \frac{1}{z} -\frac{1}{z} $, gives $$C_L^i(z,t)=\eta(z+t)\circeq\exp(\frac{i2\pi t}{24})\: \eta(z)=C_R^i(z,t)$$
and the corresponding figure
$C_L^i(z,t)$ is annihilated by $\frac{\partial }{\partial t}-\frac{\partial }{\partial z}$, while $C_R^i(z,t)$ is not, even at integer $t$.
I've scanned through quite a lot of papers containing info on the $\eta$-function yet haven't seen similar figures, but the Dedekind $\eta$ has been pretty well explored, so I was hoping someone could direct me to some references in the vast literature that might explain the geometry of such trajectories. (Obviously, a torus is evoked, but ....)
I'm aware that E. Ghys deals with similar topics in "Knots and Dynamics" (see also Site1 and Site2), but I'm not sure (yet) how to clearly connect his arguments to the above curves.
