Über einen Satz von Shl. Sternberg in der Theorie der analytischen Iterationen

Abstract

Let ℂ〚X〛=ℂ〚X ₁,...,X _n 〛 be the ring of formal power series inn indeterminates over ℂ. LetF:X→AX+B(X)=(F ⁽¹⁾(X),...,F ⁽ⁿ⁾(X))∈(ℂ〚X〛)ⁿ denote an automorphism of ℂ〚X〛 and let ϱ₁,...,ϱ _n be the eigenvalues of the linear partA ofF. We will say thatF has an analytic iteration (a. i.) if there exists a family (F _t (itX)) _t∈ℂ of automorphisms such thatF _t(X) has coefficients analytic int and such thatF ₀=X,F ₁=F,F _t+t′=F_tℴF_t′ for allt,t′∈ℂ. Let now a setΛ=(lnϱ₁,...,lnϱ _n ) of determinations of the logarithms be given. We ask if there exists an a. i. ofF such that the eigenvalues of the linear partA(t) ofF _t(X) are$e^{\ln \varrho _1 t} ,...,e^{\ln \varrho _n t}$. We will give necessary and sufficient conditions forF to have such an a. i., namely thatF is conjugate to a semicanonical formN=T ⁻¹ℴFℴT such that inN ^(k)(X) there appear at most monomialsX ^α1₁ ...X ⁿ _n $\ln \varrho _k = \sum\limits_{i = 1}^n {\alpha _i \ln } \varrho _i$. This generalizes a result of Shl.Sternberg.