Normal Forms and Linearization of Holomorphic Dilation Type Semigroups in Several Variables

In this paper we study commuting families of holomorphic mappings in Cn which form abelian semigroups with respect to their real parameter. Linearization models for holomorphic mappings are been used in the spirit of Schröder’s classical functional equation. The one-dimensional linearization models for holomorphic mappings and semigroups, based on Schröder’s and Abel’s functional equation have been studied by many mathematicians for more than a century. These models are powerful tools in investigations of asymptotic behavior of semigroups, geometric properties of holomorphic mappings and their applications to Markov’s stochastic branching processes. It turns out that solvability as well as constructions of the solution of Schröder’s or Abel’s functional equations properly, depend on the location of the so-called Denjoy–Wolff point of the given mappings or semigroups. In particular, recently many efforts were directed to the study of semigroups with a boundary Denjoy–Wolff point [4, 12, 2, 11]. Multidimensional cases are more delicate even when the Denjoy–Wolff point is inside of the underlined domain. It appears that the existence of the solution (the so-called Kœnigs’ function) of a multidimensional Schröder’s equation depends also on the resonant properties of the linear part of a given mapping (or generator), and its relation to homogeneous polynomials of higher degrees. In parallel, the study of commuting mappings (or semigroups) is of interest to many mathematicians and goes back to the classical theory of linear operators, differential equations and evolution problems. In this paper we consider, in particular, the rigidity property of two commuting semigroups. Namely, the question we study is whether those semigroups coincide whenever the linear parts of their generators at their common null point are the same. This research is part of the European Science Foundation Networking Programme HCAA. 1 2 F. BRACCI, M. ELIN, AND D. SHOIKHET Let D be a domain in C. We denote the set of holomorphic mappings on D which take values in a set Ω ⊂ C by Hol(D,Ω). For each f ∈ Hol(D,C), the Frechét derivative of f at a point z ∈ D (which is understood as a linear operator acting from C to C or n×m-matrix) will be denoted by dfz. For brevity, we write Hol(D) for Hol(D,D). The set Hol(D) is a semigroup with respect to composition operation. Definition 1. A familyS = {φt}t≥0 ⊂ Hol(D) of holomorphic self-mappings of D is called a one-parameter continuous semigroup if the following conditions are satisfied: (i) φt+s = φt ◦ φs for all s, t ≥ 0; (ii) lim t→0+ φt(z) = z for all z ∈ D. It is more or less known that condition (ii) (the right continuity of a semigroup at zero) actually implies its continuity (right and left) on all of R = [0,∞). Moreover, in this case the semigroup is differentiable on R with respect to the parameter t ≥ 0 (see [4, 12, 2, 11]). Thus, for each z ∈ D there exists the limit (1) lim t→0+ φt(z)− z t = f(z), which belongs to Hol(D,C). The mapping f ∈ Hol(D,C) defined by (1) is called the (infinitesimal) generator of S = {φt}t≥0. Furthermore, the semigroup S can be defined as a (unique) solution of the Cauchy problem: (2)    ∂φt(z) ∂t = f(φt(z)), t ≥ 0, φ0(z) = z, z ∈ D. Definition 2. We say that a semigroup {φt}t≥0 is linearizable if there is a biholomorphic mapping h ∈ Hol(D,C) and a linear semigroup {ψt}t≥0 such that {φt}t≥0 conjugates with {ψt}t≥0 by h, namely, h ◦φt = ψt ◦h for all t ≥ 0. Linearization methods for semigroups on the open unit disk in C (= C) have been studied by many mathematicians (see, for example, [14, 13, 8]). At the same time, little is known about multi-dimensional cases. For example, in [9] and [7] the problem has been studied for some special class of the so-called one-dimensional type semigroups. In this paper, we will concentrate on the case when a semigroup has a (unique) interior attractive fixed point, i.e., lim t→∞ φt(z) = τ ∈ D ⊂ C n for all z ∈ D. It is well known that this condition is equivalent to that fact that the NORMAL FORMS FOR SEMIGROUPS 3 spectrum σ(A) of the linear operator (matrix) A defined by A := dfτ lies in the open left half-plane (see [1] and [11]) and d(φt)τ = e. Usually, such semigroups are named of dilation type. Thus, for the one-dimensional case, it is possible to linearize the semigroup by solving Schröder’s functional equation: h (φt(z)) = e f ′h(z) (see, for example, [14, 12]). Remark 1. It should be noted that the latter equation involves the eigenvalue problem for the linear semigroup {Ct}t≥0 of composition operators on the space Hol(D,C) defined by Ct : h 7→ h ◦ φt. It is easy to show that the solvability of a higher dimensional analog of Schröder’s functional equation (3) h (φt(z)) = e h(z), A = dfτ , is equivalent to a generalized differential equation: (4) dhzf(z) = Ah(z). It seems that in general useful criteria (necessary and sufficient conditions) for solvability of (4) are unknown. Without loss of generality, let us assume that τ = 0. Proposition 1. Equation (3), or equivalently, (4) is solvable if and only if there is a polynomial mapping Q : C 7→ C with Q(O) = O and dQO = id, such that the limit lim t→∞ eQ(φt(z)) =: h(z), z ∈ D,