Iterations of holomorphic maps of infinite dimensional homogeneous domains (Q759966)

The result of \textit{J. Wolff} [C. R. Acad. Sci. Paris 182, 918-920 (1926); 183, 500-502 (1926)] concerning iterations of holomorphic maps on \(\Delta =\{z\in {\mathbb{C}}:| z| <1\}\) was extended by \textit{K. Fan} [Math. Z. 179, 293-298 (1982; Zbl 0465.47017)] to the case of holomorphic maps of proper contraction operators in the sense of functional calculus. Investigations of iterations of holomorphic maps of the unit balls in \({\mathbb{C}}^ N\) were carried out (by means of different methods) by \textit{B. d. MacCluer} [Mich. Math. J. 30, 97-106 (1983; Zbl 0528.32019)], \textit{Y. Kubota} [Proc. Am. Math. Soc. 88, 476-480 (1983; Zbl 0518.32016)] and \textit{G. N. Chen} [J. Math. Anal. Appl. 98, 305-313 (1984)]. In Lecture Notes Math. 364, 13-40 (1974; Zbl 0293.46049) \textit{L. A. Harris} introduced and studied a large class of infinite dimensional complex Banach spaces \({\mathfrak A}\), called \(J^*\)-algebras, whose open unit balls \({\mathfrak A}_ 0\) are bounded symmetric homogeneous domains. In the present paper regions of variability are determined and distortion theory is proved for iterations of (Fréchet-) holomorphic maps of \({\mathfrak A}_ 0\).