Two-parameter groups of formal power series

Abstract:

By ${\Omega ^F}$ we denote the group of the formal power series having the form $F(z) = \Sigma _{q = 1}^\infty {f_q}{z^q},{f_1} \ne 0$, with respect to formal composition of power series. The problem of analytic iteration leads to the study of subgroups of ${\Omega ^F}$, having the form \[ F(z,s) = \sum \limits _{q = 1}^\infty {{f_q}(s){z^q}} \] where the coefficients ${f_q}(s)$ are analytic functions of the complex parameter s, such that for any two complex numbers s and t the formal law of composition \[ F[F(z,s),t] = F(z,s + t)\] is valid [6], [8]. The purpose of the present paper is to study similar two-parameter subgroups of ${\Omega ^F}$. In §1 r-parameter analytic subgroups of ${\Omega ^F}$ are defined, as well as other concepts connected with the problem. In §2 the importance of two-parameter subgroups is emphasized. It is shown that the number of parameters of analytic subgroups of ${\Omega ^F}$ can always be reduced to two at most. The existence of a countable number of classes of the two-parameter subgroups of ${\Omega ^F}$ is shown. §3 gives the explicit form of the coefficients ${f_q}({a^1},{a^2})$ of a two-parameter subgroup of ${\Omega ^F}$: \[ F(z,{a^1},{a^2}) = \sum \limits _{q = 1}^\infty {{f_q}({a^1},{a^2}){z^q}} .\] In §4 the existence of canonical representations for two-parameter analytic subgroups of ${\Omega ^F}$ is proven, and it is shown that every two-parameter analytic subgroup of ${\Omega ^F}$ is globally isomorphic to one of the groups \[ {H_n}(z,{a^1},{a^2}) = (1 + {a^1})z/{(1 + {a^2}{z^n})^{1/n}},\quad n = 1,2, \ldots \] (no two of which are globally isomorphic to each other).