ACTA issues

Iterates of a compact holomorphic map on a finite rank homogeneous ball

M. Mackey, P. Mellon

Acta Sci. Math. (Szeged) 85:1-2(2019), 203-214

Abstract. We study iterates, $f^n$, of a fixed-point free compact holomorphic map $f\colon B\rightarrow B$ where $B$ is the open unit ball of any $JB^*$-triple of finite rank. These spaces include $L(H,K)$, $H,K$ Hilbert, dim$(H)$ arbitrary, dim$(K)< \infty $, or any classical Cartan factor or $C^*$-algebra of finite rank. Apart from the Hilbert ball, the sequence of iterates $(f^n)_n$ does not generally converge (locally uniformly on $B$) and little is known of accumulation points. We present a short proof of a Wolff theorem for $B$ and establish key properties of the resulting $f$-invariant subdomains. We define a concept of closed convex holomorphic hull, $\mathop{\rm Ch}(x)$, for $x \in\partial B$ and prove the following. There is a unique tripotent $u$ in $\partial B$ such that all constant subsequential limits of $(f^n)_n$ lie in $\mathop{\rm Ch}(u)$. As a consequence we also get a short proof of the classical Hilbert ball results.

DOI: 10.14232/actasm-018-518-z

AMS Subject Classification (1991): 47H10, 32M15; 32H50, 58C10

Keyword(s): iteration, bounded symmetric domain, Denjoy--Wolff theorem

Received February 8, 2018 and in final form March 8, 2018. (Registered under 18/2018.)