  We study the complex Dulac map for a holomorphic foliation of the complex
plane, near a non-degenerate singularity (both eigenvalues of the linearization
are nonzero) with two separatrices. Following the well-known results of Y.
Il'yashenko we provide a geometric approach allowing to study the whole maximal
domain of (geometric) definition of the Dulac map. In particular its topology
and the regularity of its boundary are completely described. We also study the
order of magnitude of the first non-trivial term of its asymptotic expansion
and show how to compute it using path integrals supported in the leaves of the
linearized foliation. Explicit bounds on the remainder are given. We perform
similarly the study of the Dulac time spent around the singularity. All results
are formulated in a unified framework taking no heed to the usual dynamical
discrimination (i.e. no matter whether the singularity is formally orbitally
linearizable or not and regardless of the arithmetic of the eigenvalues ratio).
