# Whiskered KAM Tori of Conformally Symplectic Systems

## TL;DR

This paper proves the existence of whiskered tori in conformally symplectic systems with a constructive algorithm, extending KAM theory to dissipative settings with specific invariance and splitting conditions.

## Contribution

It establishes the existence of whiskered tori in conformally symplectic maps with an explicit iterative method under non-degeneracy and approximate invariance assumptions.

## Key findings

- Existence of invariant whiskered tori in conformally symplectic systems.
- Development of an efficient algorithm for constructing such tori.
- Validation of theoretical conditions for persistence of invariant structures.

## 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].