Hypertranscendency of conjugacies in complex dynamics (Q1891208)

The dynamics of rational functions near fixed points are closely connected to the functional equations of Schröder, Böttcher, and Abel. We show that the solutions of these equations are hypertranscendental except in some cases, where they are given in terms of exponential, trigonometric, and elliptic functions. This generalizes earlier work of \textit{J. F. Ritt} [Math. Ann. 95, 671-682 (1926; JFM 52.0321.04)], \textit{F. W. Carroll} [Mich. Math. J. 32, 47-57 (1985; Zbl 0574.30028)], and \textit{P. Borwein} [Proc. Am. Math. Soc. 107, 215-221 (1989; Zbl 0674.39005)].