**Author**
: Guido Gentile

**Title:** *
Degenerate lower-dimensional tori under the Bryuno condition
*

**Abstract**:
We study the problem of conservation of maximal and lower-dimensional
invariant tori for analytic convex quasi-integrable Hamiltonian systems.
In the absence of perturbation the lower-dimensional tori are degenerate,
in the sense that the normal frequencies vanish, so that the tori are
neither elliptic nor hyperbolic. We show that if the perturbation
parameter is small enough, for a large measure subset of any resonant
submanifold of the action variable space, under some generic
non-degeneracy conditions on the perturbation function,
there are lower-dimensional tori which are conserved.
They are characterised by rotation vectors satisfying some generalised
Bryuno conditions involving also the normal frequencies.
We also show that, again under some generic assumptions on the
perturbation, any torus with fixed rotation vector satisfying
the Bryuno condition is conserved for most values of the perturbation
parameter in an interval small enough around the origin.
According to the sign of the normal frequencies and of the
perturbation parameter the torus
becomes either hyperbolic or elliptic or of mixed type.

**Keywords**:
KAM theory, Divergent Series, Lindstedt Series,
Lower-dimensional elliptic tory, Degeneracy, Bryuno condition,
Resummations, Renormalization Group, Quantum Field Theory

Guido Gentile

Dipartimento di Matematica, Università
di Roma Tre

L. S. Leonardo Murialdo

00100 Roma, Italia

tel +39-06-54-888-226

e-mail: gentile@mat.uniroma3.it