01-447 David DeLatte, Todor Gramchev

Biholomorphic maps with linear parts having Jordan blocks: linearization and resonance type phenomena (73K, AMS-TEX) Dec 4, 01

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Abstract. We study the linearizability of biholomorphic maps of $\C^n$ fixing the origin when the Jacobian matrix admits a nontrivial Jordan block. Our main result proves convergence of the linearizing transformation of maps for which the Jordan part of the spectrum lies inside the unit circle and the spectrum satisfies a R\"ussmann-type diophantine condition. Degeneracy of the Jordan block - different geometric and algebraic multiplicity - is allowed. The key to the proof is the decoupling of the homological equation into the Siegel part and Poincar\'e part. In higher dimensions ($n>3$) new inhomogeneous diophantine conditions also appear. We show that quasi-resonance phenomena occur and that when a nontrivial Jordan block is presen the homological equation cannot be solved in general due to the accumulated effects of small divisors. In the purely hyperbolic case Jordan blocks are an obstruction to holomorphic linearization - even under diophantine conditions.

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