visualizing functions invariant (or almost) under modular transformation In the spirit of Möbius Transformations Revealed, I'd like to make a pair of movies depicting how Klein's absolute invariant $j(\tau)$ and the Dedekind eta function $\eta(\tau)$ transform when $\tau\mapsto\frac{a\tau+b}{c\tau+d}$, but I'm sort of stuck because the modular group $\Gamma$ is a discrete group. I could linearly interpolate, but I suspect there's a better approach involving $\mathbf{SL}(2,\mathbb{R})$. 
edit: video generated: Klein's j-Invariant $j(τ)$ under $τ→-1/τ$ in $\mathbf{SL}(2,\mathbb{R})$
 A: The matrices
$$S=\begin{pmatrix}0&-1\\1&~~\,0\end{pmatrix}, \qquad T=\begin{pmatrix}1&1\\0&1\end{pmatrix}$$
generate ${\rm SL}_2({\bf Z})$. As elements in the lie group ${\rm SL}_2({\bf R})$ they each arise in a unique one-parameter subgroup, which is useful for our purposes because it is a smooth curve. In particular, to animate the action of $S$ on $z\mapsto f(z)$ (which yields $(Sf):z\mapsto f(S^{-1}z)$), each frame in the animation may be chosen to depict $(S_tf)$ as $t$ varies in $[0,1]$, where we write $S_t=\exp(t\log S)$. The exact same idea applies to $T$: simply animate $T_tf$ with $T_t=\exp(t\log T)$. Here are the logarithms:
$$\log \begin{pmatrix}0&-1\\1&~~\,0\end{pmatrix}=\frac{\pi}{2}\begin{pmatrix}0&-1\\1&~~\,0\end{pmatrix}, \qquad \log \begin{pmatrix}1&1\\0&1\end{pmatrix} = \begin{pmatrix}0&1\\0&0\end{pmatrix}.$$
Both of these are cited over at Wikipedia. Note that the latter matrix when animated will in fact pretty much be linear translation, but the former matrix yields "rotation" on the upper halfplane.
The exponentials are given exactly by
$$S_t=\begin{pmatrix}\cos(\pi t/2) & -\sin(\pi t/2) \\ \sin(\pi t/2) & ~~~\cos(\pi t/2)\end{pmatrix}, \qquad T_t=\begin{pmatrix}1&t\\0&1\end{pmatrix}. $$
To animate a word's action, apply each letter one at a time from right to left to a function.
