Whiskered KAM Tori of Conformally Symplectic Systems

We investigate the existence of whiskered tori in some dissipative systems, called conformally symplectic systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family fµ of conformally symplectic maps which depends on a drift parameter µ. We fix a Diophantine frequency of the torus and we assume to have a drift µ0 and an embedding of the torus K0, which satisfy approximately the invariance equation fµ0 ◦ K0 = K0 ◦ Tω (where Tω denotes the shift by ω). We also assume to have a splitting of the tangent space at the range of K0 into three bundles. We assume that the bundles are approximately invariant under D fµ0 and the derivative satisfies some rate conditions. Under suitable nondegeneracy conditions, we prove that there exist µ∞, K∞ invariant under fµ∞ , close to the original ones, and a splitting which is invariant under D fµ∞ . The proof provides an efficient algorithm to construct whiskered tori. Full details of the statements and proofs are given in [10].