David DeLatte, Todor Gramchev
Biholomorphic maps with linear parts having Jordan blocks:
linearization and resonance type phenomena
(73K, AMS-TEX)
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.