Invariant distances and metrics in complex analysis–-revisited

A Julia's Lemma for the symmetrized bidisc 𝔾2

“…G 2 with the following properties: To be more precise on point (a) of the previous proposition, one can actually prove (see, e.g. [1]) the following proposition.…”

“…A first link between complex geodesics in G 2 and holomorphic automorphisms of G 2 is given in the following lemma (see [1]). …”

“…We finally recall that for any two points (s 1 , [1, p. 37]) and that a complete description of complex geodesics in G 2 is given in [1]; it essentially depends on the possible intersection of the complex geodesic with the royal variety S, namely …”

“…Clearly @G 2 & G 2 (the closure of G 2 in C 2 ), but notice that the condition js À spj þ j pj 2 1 does not imply that ðs, pÞ 2 G 2 , since ð5=2, 1Þ 6 2 G 2 . The previous counterexample is made more precise in the following Proposition [1].…”

“…Let G 2 be the symmetrized bidisc, 1 that is the image of the bidisc D 2 ¼ {(z 1 , z 2 ) 2 C 2 : jz 1 j51, jz 2 j51} under ¼ ( 1 , 2 ), with j : D 2 ! D, j ¼ 1, 2, and 1 (z 1 , z 2 ) ¼ z 1 þ z 2 :¼ s, 2 (z 1 , z 2 ) ¼ z 1 Á z 2 :¼ p. The symmetrized bidisc can be considered as a special case of the symmetrized n-disc (n !…”