Parametric rigidity of real families of conformal diffeomorphisms tangent to x → - x

We prove that one-parameter families of real germs of conformal diffeomorphisms tangent to the involution x bar right arrow -x are rigid in the parameter. We establish a connection between the dynamics in the Poincare and Siegel domains. Although repeatedly employed in the literature, the dynamics in the Siegel domain does not explain the intrinsic real properties of these germs. Rather, these properties are fully elucidated in the Poincare domain, where the fixed points are linearizable. However, a detailed study of the dynamics in the Siegel domain is of crucial importance. We relate both points of view on the intersection of the Siegel normalization domains.