Dynamics of Biholomorphic Self-Maps on Bounded Symmetric Domains

Abstract

Let $g$ be a fixed-point free biholomorphic self-map of a bounded symmetric domain $B$. It is known that the sequence of iterates $(g^n)$ may not always converge locally uniformly on $B$ even, for example, if $B$ is an infinite dimensional Hilbert ball. However, $g=g_a\circ T$, for a linear isometry $T$, $a=g(0)$ and a transvection $g_a$, and we show that it is possible to determine the dynamics of $g_a$. We prove that the sequence of iterates $(g_a^n)$ converges locally uniformly on $B$ if, and only if, $a$ is regular, in which case, the limit is a holomorphic map of $B$ onto a boundary component (surprisingly though, generally not the boundary component of $\frac{a}{\|a\|}$). We prove $(g_a^n)$ converges to a constant for all non-zero $a$ if, and only if, $B$ is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.