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Diagonalization of nondiagonalizable discrete holomorphic dynamical systems
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 122, Number 4, August 2000
- pp. 757-781
- 10.1353/ajm.2000.0024
- Article
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We shall describe a canonical procedure to associate to any (germ of) holomorphic self-map F of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]n fixing the origin so that dFO is invertible and nondiagonalizable an n-dimensional complex manifold M, a holomorphic map π: M → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]n, a point e ∈ M and a (germ of) holomorphic self-map [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] of M such that: π restricted to M\π-1(O is a biholomorphism between M\π-1(O) and [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]n\{O}; π ○ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /] = F ○ π; and e is a fixed point of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] such that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] is diagonalizable. Furthermore, we shall use this construction to describe the local dynamics of such an F nearby the origin when sp (dFO) = {1}.