Contracting rigid germs in higher dimensions
Following Favre, we define a holomorphic germ to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of and its linear action on the fundamental group of the complement of the critical set.