Holomorphic mappings and their fixed points on Spin Factors

TL;DR

This paper investigates fixed points of biholomorphic automorphisms and the dynamics of holomorphic maps on infinite dimensional spin factors, revealing conditions for fixed points and describing the structure of iterates' accumulation points.

Contribution

It introduces conditions for fixed points of automorphisms on spin factors and characterizes the target set of iterates of holomorphic maps, advancing understanding of infinite dimensional complex geometry.

Findings

Spin factors are outliers among Banach spaces with homogeneous open unit balls.

Explicit fixed points on the boundary are constructed under certain conditions.

The target set of iterates of holomorphic maps lies on the boundary of a unique bidisc.

Abstract

In this paper we study holomorphic properties of infinite dimensional spin factors. Among the infinite dimensional Banach spaces with homogeneous open unit balls, we show that the spin factors are natural outlier spaces in which to ask the question (as was proved in the early 1970s for Hilbert spaces): Do biholomorphic automorphisms $g$ of the open unit ball $B$ have fixed points in $\overline B$? In this paper, for infinite dimensional spin factors, we provide reasonable conditions on $g$ that allow us to explicitly construct fixed points of $g$ lying on $\partial B$. En route, we also prove that every spin factor has the density property. In another direction, we focus on (compact) holomorphic maps $f:B\rightarrow B$, having no fixed point in $B$ and examine the sequence of iterates $(f^n)$. As $(f^n)$ does not generally converge, we instead trace the target set $T(f)$ of $f$, that is, the images of all accumulation points of $(f^n)_n$, for any topology finer than the topology of pointwise convergence on B. We prove for a spin factor that $T(f)$ lies on the boundary of a single bidisc unique to $f$.