MPEJ Volume 8, No. 2, 27 pp.
Received: Dec 4, 2001. Accepted: Feb 14, 2002.

DeLatte D., Gramchev T.
Biholomorphic maps with linear parts having Jordan blocks:
linearization and resonance type phenomena

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 present
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.

http://www.maia.ub.es/mpej/Vol/8/2.ps
http://www.ma.utexas.edu/mpej/Vol/8/2.ps
http://mpej.unige.ch/mpej/Vol/8/2.ps