On horospheres and holomorphic endomorfisms of the Siegel disc
The concept of horocycle and horosphere in the unit disc of \({\mathbb{C}}\) have been introduced by Poincaré. Similar notions of horospheres can be defined in the unit ball \(B_ n\) (for the euclidean norm) of \({\mathbb{C}}^ n\). The horospheres of \(B_ n\) are characterized in terms of the Kobayashi distance, which plays, in this case, the role of the Poincaré distance [see \textit{P. Yang}, ''Holomorphic curves and boundary regularity of biholomorphic maps of pseudoconvex domains'', unpublished.]. One of the most important results about horospheres is the classical Julia's lemma. \textit{P. C. Yang} [see the above cited paper; and \textit{L. Nirenberg}, \textit{S. Webster} and \textit{P.C. Yang}, Commun. Pure Appl. Math. 33, 305-338 (1980; Zbl 0436.32018)] has extended these concepts and Julia's lemma to strictly pseudo-convex domains of \({\mathbb{C}}^ n\), with smooth boundary. In this paper, we shall introduce the notions of horosphere and horocycle in the Siegel disc \({\mathcal E}\). We characterise the Šilov boundary of horospheres in terms of the Kobayashi distance and we establish an extension of Julia's lemma. In the last part of the article it is proved that if F is a holomorphic endomorphism of \({\mathcal E}\), which behaves ''regularly'' on a horocycle and near a boundary point of \({\mathcal E}\), then \(F\in Aut({\mathcal E})\).
scientific article; zbMATH DE number 3925281
scientific article