A Julia's Lemma for the symmetrized bidisc 𝔾2

Abstract

In this article we provide a result that may be considered as an extension of the Julia's Lemma to the case of holomorphic self-maps in the intriguing domain known as the symmetrized bidisc. Julia's Lemma is a classical result for holomorphic self-maps in the (poly)disc, and it turns out to be one of the starting points for the study of iterates of holomorphic self-maps. In the setting of the symmetrized bidisc, this kind of study towards a description of behaviour of iterates of holomorphic self-maps is of great interest and is partially still under investigation. The techniques involved in this article seem to be very well suited for the case of symmetric bidisc and resemble most of the analogous properties in the case of the polydisc.