A local form of automorphisms of the spectral unit ball

TL;DR

This paper demonstrates that automorphisms of the spectral unit ball can be approximated by simpler transformations and that all such automorphisms are locally conjugate to holomorphic maps, advancing understanding of their structure.

Contribution

It proves the density of triangular and diagonal conjugations in the automorphism group and shows all automorphisms are locally holomorphic conjugations.

Findings

Triangular and diagonal conjugations are dense in automorphisms.

Every automorphism is a local holomorphic conjugation.

Provides new insights into the structure of automorphisms of the spectral unit ball.

Abstract

We show that the group generated by by triangular and diagonal conjugations is dense in $\aut(\Omega_2)$ (in the local-uniform topology). Moreover, it is shown that any automorphism of $\Omega_2$ is a local holomorphic conjugation.