Diagonalization of non-diagonalizable discrete holomorphic dynamical systems
Abstract
We describe a canonical procedure for associating to any (germ of) holomorphic self-map f of Cn fixing the origin such that dfO is invertible and non-diagonalizable an n-dimensional complex manifold M, a holomorphic map p from M to Cn, a point e in M and a (germ of) holomorphic self-map F of M so that: p restricted to the complement of p-1(O) is a biholomorphism between this complement and Cn minus the origin; p semiconjugates f and F; and e is a fixed point of F such that dFe is diagonalizable. Furthermore, we use this construction to describe the local dynamics of such an f nearby the origin when the only eigenvalue of dfO is 1.
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