Paper IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round,line join=round]
2 \draw[black!45] (2.70000,2.50000) -- (4.70000,2.50000);
3 \draw[fill=black] (3.30000,2.50000) circle (0.055);
4 \draw[fill=black] (4.50000,2.50000) circle (0.055);
5 \draw[black!45] (2.70000,2.50000) -- (2.70000,4.50000);
6 \draw[fill=black] (2.70000,3.10000) circle (0.055);
7 \draw[fill=black] (2.70000,4.30000) circle (0.055);
8 \draw[black!45] (2.70000,2.50000) -- (0.70000,2.50000);
9 \draw[fill=black] (2.10000,2.50000) circle (0.055);
10 \draw[fill=black] (0.90000,2.50000) circle (0.055);
11 \draw[black!45] (2.70000,2.50000) -- (2.70000,0.50000);
12 \draw[fill=black] (2.70000,1.90000) circle (0.055);
13 \draw[fill=black] (2.70000,0.70000) circle (0.055);
14 \node[] at (2.70000,5.00000) {$u=z/(\sqrt2\pi)$};
15 \node[] at (2.70000,-0.30000) {$u=\tfrac12 i^k,\ \tfrac32 i^k$};
16 \draw[->,line width=.9pt] (5.30000,2.50000) -- (7.00000,2.50000);
17 \node[] at (6.15000,3.00000) {$v=u^4$};
18 \draw[->] (7.50000,2.50000) -- (12.80000,2.50000);
19 \draw[fill=black] (7.65000,2.50000) circle (0.09);
20 \node[] at (7.65000,1.95000) {$1/16$};
21 \node[] at (7.65000,3.10000) {one zero};
22 \draw[fill=black] (12.10000,2.50000) circle (0.09);
23 \node[] at (12.10000,1.95000) {$81/16$};
24 \node[] at (12.10000,3.10000) {one zero};
25 \node[] at (10.20000,5.00000) {$v=w/(4\pi^4)$};
26 \node[] at (10.20000,-0.30000) {four points per orbit become one};
27 \end{tikzpicture}

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