Open Access Open Access  Restricted Access Subscription Access

Dynamics of Biholomorphic Self-Maps on Bounded Symmetric Domains

P. Mellon


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.

Full Text:




  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.

ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library