Dynamics of the arithmetic-geometric mean (Q809441)

The author considers Gauss's arithmetic-geometric mean (agm) in its inhomogeneous form. Instead of \((a,b)\to (a',b')\), he considers \(z\to z'\) where \(z=a/b\) and \(z'=a'/b'\). This leads to \(z'=(1+z)/2\sqrt{z},\) a special (agm) case of the polynomial correspondence that is quadratic in two variables. So the type of correspondence can be viewed in terms of dynamics. The phenomenal (agm) convergence is merely the strong attractor at \(z=1\). The author shows that the nature of the critical points \((z=1,- 1,0,\infty)\) will characterize agm to within linear maps. He uses Gauss's theta-functions to parametrize the upper-half plane so that the action of agm will transfer to elements of \(\Gamma_ 2(4)\). In this way the agm sequence is a linear-fractional action of integral matrices of determinant 2. The author also considers the relation of periodic orbits with class number of the discriminant of the eigenvalue equation. Finally, he lists other types of critical correspondences distinct from the agm, and he classifies them according to dynamic behavior. He cites as references, \textit{J. Borwein} and \textit{P. Borwein} [Pi and the AGM (1987; Zbl 0611.10001)], \textit{D. Cox} [Enseign. Math., II. Sér. 30, 275-330 (1984; Zbl 0583.33002)] and \textit{J. J. Gray} [Expo. Math. 2, 97- 130 (1984; Zbl 0539.01008)]. [Reviewer's note: The work of \textit{J. Todd}, Numer. Math. 54, 1-18 (1988; Zbl 0668.40001), should also be cited in reference to periodic orbits.]