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We prove that a topological homeomorphism conjugating two generic 1-parameterunfoldings of 1-variable complex analytic resonant diffeomorphisms isholomorphic or anti-holomorphic by restriction to the unperturbed parameter. Weprovide examples that show that the genericity hypothesis is necessary.Moreover we characterize the possible behavior of conjugacies for theunperturbed parameter in the general case. In particular they are always realanalytic outside of the origin.  We describe the structure of the limits of orbits when we approach theunperturbed parameter. The proof of the rigidity results is based on the studyof the action of a topological conjugation on the limits of orbits.

Topological rigidity of unfoldings of resonant diffeomorphisms