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Topologies on the set of iterates of a holomorphic function in infinite dimensions

Volume 70 / 2022

M. Mackey, P. Mellon Bulletin Polish Acad. Sci. Math. 70 (2022), 151-158 MSC: Primary 46E50; Secondary 46G20, 32H50. DOI: 10.4064/ba220731-22-12 Published online: 18 January 2023


Let $f:B\to B$ be a compact holomorphic map on the open unit ball $B$ of a complex Banach space $Z$ in possibly infinite dimensions, where $f$ compact means $f(B)$ is relatively compact. The sequence of iterates $(f^n)_n$ of $f$ (where $f^n:=f \circ f^{n-1}$, $f^1:=f$) is of much interest and, since it generally does not converge, the set of all its subsequential limits for a particular topology have been studied instead.

We prove that the pointwise limit of any subsequence of $(f^n)_n$ is itself a holomorphic function. We show, in fact, that on the set of iterates $\{f^n:n \in \mathbb N\}$ the topology of pointwise convergence on $B$ coincides with any finer topology on the space $H(B,Z)$ of holomorphic functions from $B$ to $Z$. In particular, it coincides with both the compact-open topology and the topology of local uniform convergence on $B$. Despite the fact that these topologies are not first countable, we prove that the set of accumulation points of $(f^n)_n$ coincides with the set of all its subsequential limits.


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