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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 $C_R(z,t)$

alt text

$C_L(z,t)$ 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} $, 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

alt text

$C_L^i(z,t)$ 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.

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Hi Tom--are these figures special to the $\eta$ function, or do similar figures arise if one makes similar parametric plots with some random analytic function (say a polynomial) instead? – Daniel Litt Dec 20 '12 at 6:17
Hi Dan--the results above depend on $\eta$ being a modular form. I don't see how to get the necessary periodicity with any polynomial under shear transformations. Scan my notes on Infinigens at my little "arxiv website" for details. – Tom Copeland Dec 20 '12 at 10:31
We're essentially looking at solns. around $z=0$ and infinity of the eigenfunction eqn. $\exp(t\cdot z^2 \frac{\partial }{\partial z}) f(z)=f[z/(-tz+1)]=\epsilon(t) f(z)$ for some $t$. The simplest case being $f(z)=\exp(-\lambda/z)$. – Tom Copeland Dec 20 '12 at 13:42
I wonder if the first curve could be lifted off the plane onto a double torus with one torus nested inside the other but sharing its circle of max radius with that of the other torus. – Tom Copeland Dec 21 '12 at 0:37

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