A Dedekind Eta trajectory / horocyclic flow (Reference request) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:47:46Z http://mathoverflow.net/feeds/question/116561 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116561/a-dedekind-eta-trajectory-horocyclic-flow-reference-request A Dedekind Eta trajectory / horocyclic flow (Reference request) Tom Copeland 2012-12-17T03:49:22Z 2012-12-20T03:38:45Z <p>I've been exploring the composition of essentially the Dedekind $\eta$-function with parabolic Möbius transformations, </p> <p>$$C_L(z,t)=\left(\frac{z}{-tz+1}\right)^{\frac{1}{2}}\eta\left(\frac{z}{-tz+1}\right)\circeq\exp(\frac{i2\pi t}{24})\: z^{\frac{1}{2}}\:\eta(z)=C_R(z,t),$$</p> <p>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 $-12 \leq t \leq 12$ :</p> <p>$xL(t)=Real[C_L(z,t)]\:\:\:$ and $\:\:\:yL(t)=Imag[C_L(z,t)]$ and analogously for $C_R(z,t)$ </p> <p><img src="http://tcjpn.files.wordpress.com/2012/12/img0332.png" alt="alt text"></p> <p>$C_L(z,t)$ is annihilated by $\frac{\partial }{\partial t}-z^2\frac{\partial }{\partial z}$, while $C_R(z,t)$ is not, even at integer $t$.</p> <p>Letting $z \mapsto -\frac{1}{z}$, gives $$C_L^i(z,t)=\eta(z+t)\circeq\exp(\frac{i2\pi t}{24})\: \eta(z)=C_R^i(z,t)$$</p> <p>and the corresponding figure</p> <p><img src="http://tcjpn.files.wordpress.com/2012/12/img00681.png" alt="alt text"></p> <p>$C_L^i(z,t)$ is annihilated by $\frac{\partial }{\partial t}-\frac{\partial }{\partial z}$, while $C_R^i(z,t)$ is not, even at integer $t$.</p> <p><strong>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 ....)</strong> </p> <p>I'm aware that E. Ghys deals with similar topics in "<a href="http://www.umpa.ens-lyon.fr/~ghys/articles/icm.pdf" rel="nofollow">Knots and Dynamics</a>" (see also <a href="http://www.ams.org/samplings/feature-column/fcarc-lorenz" rel="nofollow">Site1</a> and <a href="http://terrytao.wordpress.com/2007/08/03/2006-icm-etienne-ghys-knots-and-dynamics/" rel="nofollow">Site2</a>), but I'm not sure (yet) how to clearly connect his arguments to the above curves.</p>