Existence of whiskered KAM tori of conformally symplectic systems

Abstract

We study the existence of whiskered tori in a family $f_\mu$ of conformally symplectic maps depending on parameters $\mu$. Whiskered tori are tori on which the motion is a rotation, but they have as many expanding/contracting directions as allowed by the preservation of the geometric structure. Our main result is formulated in an "a-posteriori" format. We fix $\omega$ satisfying Diophantine conditions. We assume that we are given 1) a value of the parameter $\mu_0$, 2) an embedding of the torus $K_0$ into the phase space, approximately invariant under $f_{\mu_0}$ in the sense that $f_{\mu_0} \circ K_0 - K_0 \circ T_\omega$ is small, 3) a splitting of the tangent space at the range of $K_0$, into three bundles which are approximately invariant under $D f_{\mu_0}$ and such that the derivative satisfies "rate conditions" on each of the components. Then, if some non-degeneracy conditions are satisfied, we show that there is another parameter $\mu_\infty$, an embedding $K_\infty$ and splittings close to the original ones which are invariant under $f_{\mu_\infty}$. We also bound $|\mu_\infty - \mu_0|$, $\|K_\infty - K_0 \|$ and the distance of the initial and final splittings in terms of the initial error. The proof of the main theorem consists in describing an iterative process that takes advantage of cancellations coming from the geometry. Then, we show that the process converges to a true solution when started from an approximate enough solution. The iterative process leads to an efficient algorithm that is quite practical to implement. As an application, we study the singular problem of effects of small dissipation on whiskered tori. We develop formal expansions in the perturbative parameter and use them as input for the a-posteriori theorem. This allows to obtain lower bounds for the domain of analyticity of the tori as function of the perturbative parameter.