A Parameterization Method for Lagrangian Tori of Exact Symplectic Maps of $R^2r$

Abstract

We are concerned with analytic exact symplectic maps of ${\mathbb R}^{2r}$ endowed with the standard symplectic form. We study the existence of a real analytic torus of dimension $r$, invariant by the map and carrying quasi-periodic motion with a prefixed Diophantine rotation vector. Therefore, this torus is a Lagrangian manifold. We address the problem by the parameterization method in KAM theory. The main aspect of our approach is that we do not look for the parameterization of the torus as a solution of the corresponding invariance equation. Instead, we consider a set of three equations that, all together, are equivalent to the invariance equation. These equations arise from the geometric and dynamical properties of the map and the torus. Suppose that an approximate solution of these equations is known and that a suitable nondegeneracy (twist) condition is satisfied. Then, this system of equations is solved by a quasi-Newton-like method, provided that the initial error is sufficiently small. By “quasi-Newton-like” we mean that the convergence is almost quadratic but that at each iteration we have to solve a nonlinear equation. Although it is straightforward to build a quasi-Newton method for the selected set of equations, proceeding in this way we improve the convergence condition. The selected definition of error reflects the level at which the error associated with each of these three equations contributes to the total error. The map is not required to be close to integrable or expressed in action-angle variables. Suppose the map is $\varepsilon$-close to an integrable one, and consider the portion of the phase space not filled up by Lagrangian invariant tori of the map. Then, the upper bound for the Lebesgue measure of this set that we may predict from the result is of ${\mathcal O}(\varepsilon^{1/2})$. In light of the classical KAM theory for exact symplectic maps, an upper bound of ${\mathcal O}(\varepsilon^{1/2})$ for this measure is the expected estimate. The result also has some implications for finitely differentiable maps.