Whiskered KAM Tori of Conformally Symplectic Systems

Abstract

We investigate the existence of whiskered tori in some dissipative systems, called \sl conformally symplectic \rm systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family $f_\mu$ of conformally symplectic maps which depend on a drift parameter $\mu$. We fix a Diophantine frequency of the torus and we assume to have a drift $\mu_0$ and an embedding of the torus $K_0$, which satisfy approximately the invariance equation $f_{\mu_0} \circ K_0 - K_0 \circ T_\omega$ (where $T_\omega$ denotes the shift by $\omega$). We also assume to have a splitting of the tangent space at the range of $K_0$ into three bundles. We assume that the bundles are approximately invariant under $D f_{\mu_0}$ and that the derivative satisfies some "rate conditions". Under suitable non-degeneracy conditions, we prove that there exists $\mu_\infty$, $K_\infty$ and splittings, close to the original ones, invariant under $f_{\mu_\infty}$. The proof provides an efficient algorithm to construct whiskered tori. Full details of the statements and proofs are given in [CCdlL18].