12 - Biholomorphic germs

Summary

This chapter is about the reversible elements in the group G of invertible biholomorphic germs in one variable and some of its subgroups. The theory of reversibility for formally-reversible formal power series in one variable has already been dealt with in Chapter 10. In the present chapter, we shall denote the group of these formal series by G. The group G is a subgroup of G, which implies that reversible biholomorphic germs are formally reversible.

A priori, biholomorphic conjugacy is a much finer relation than formal conjugacy, so one expects that the formal conjugacy class of an element of G will split into many distinct biholomorphic conjugacy classes. This is often the case, and indeed we shall see (in Section 12.6) that there exist germs that are formally reversible, but not biholomorphically reversible.

Let us discuss the groups to be studied in more detail. A germ at 0 is an equivalence class of functions under the relation that regards two functions as equivalent if they agree on some neighbourhood of 0. Let S denote the set of those invertible complex holomorphic maps defined on a neighbourhood of 0 that fix 0. The group G of biholomorphic germs consists of the equivalence classes of S under the equivalence relation just described. Thus an element of G is represented by some function f, holomorphic on some neighbourhood of 0 (which depends on f) with f (0) = 0 and f′(0) ≠ 0. Two such functions represent the same germ if they agree on some neighbourhood of 0. The group operation is composition and the identity is the germ of the identity function z ↦ z, which we denote by 1.

Next we introduce notation for some important subgroups of G and for some quantities that were used in Chapter 10 to distinguish the conjugacy classes of formal power series. Each element of G can be represented by a convergent complex power series with no constant term and with a nonzero z coefficient.