  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.
