Denjoy–Wolff theory for finite-dimensional bounded symmetric domains

Abstract

Let \(B\) be a finite-dimensional bounded symmetric domain and \(f:B\rightarrow B\) be a holomorphic map having no fixed point in \(B\). For subsequential limits, \(g\), of \((f^n)\), we establish conditions, in terms of the Wolff point, \(\xi \), of \(f\), on which boundary components of \(B\) can contain \(g(B)\). We extend Herve’s 1954 theorem on the bidisc to any finite product of bounded symmetric domains, namely if \(B=B_1\times \cdots \times B_n\) and \(\xi =(\xi _1,\ldots ,\xi _n)\) then there exists \(d=(d_1,\ldots , d_n) \in \partial B\), satisfying \( \overline{K_{d_i}} \cap \overline{K_{\xi _i}}\ne \emptyset ,\) such that $$\begin{aligned} \pi _i(g(B))\subseteq d_i+B_0(d_i), \end{aligned}$$ where \(K_x\) denotes the affine boundary component of \(x\), \(\pi _i\) is projection on the \(i\)th coordinate and \(B_0(d_i)\) is a bounded symmetric subdomain of \(B_i\). This simplifies if \(\xi _i\) is extreme, and even more so if \(B_i\) is a Hilbert ball.