**Author:**
Guido Gentile

**Title:** *
Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers
*

**Abstract: **
Since Moser's seminal work it is well known that the invariant curves of smooth nearly
integrable twist maps of the cylinder with Diophantine rotation number are preserved under
perturbation. In this paper we show that, in the analytic class, the result extends to Bryuno
rotation numbers. First, we will show that the series expansion for the invariant curves
in powers of the perturbation parameter can be formally defined, then we shall prove that
the series converges absolutely in a neighbourhood of the origin. This will be achieved
using multiscale analysis and renormalisation group techniques to express the coefficients
of the series as sums of values which are represented graphically as tree diagrams and
then exploit cancellations between terms contributing to the same perturbation order. As
a byproduct we shall see that, when perturbing linear maps, the series expansion for an
analytic invariant curve converges for all perturbations if and only if the corresponding
rotation number satises the Bryuno condition.

**Keywords:**
Exact symplectic maps; twist maps; invariant curves; KAM theory;
Bryuno condition; renormalisation group; trees.

Guido Gentile

Dipartimento di Matematica e Fisica

Università Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it