Contracting rigid germs in higher dimensions

Abstract

30 pages, 0 figuresInternational audienceFollowing Favre, we define a holomorphic germ f:(C^d,0) -> (C^d,0) 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 f and its linear action on the fundamental group of the complement of the critical set

Böttcher coordinates at superattracting fixed points of holomorphic skew products

“…The germ f can be written as (z δ (1 + ε(z, w)), q(z, w)), where ε converges to 0 as z and w tend to 0. Moreover, Theorem 1.3 in [9] induces the following. Proposition 6.5.…”

“…Proof. We briefly review the proof in [9] following a slightly different presentation. Define Then φ n is well-defined on a small neighborhood of the origin, and…”

Classification of one-dimensional superattracting germs in positive characteristic

“…Remark 7.2. Using similar techniques, it is possible to extend some of the results in [Rug13] over (normed) fields K of characteristic p > 0. In particular, Theorem 2.7 in op.cit.…”

“…The proof of Theorem C is analogous to the one working in the complex setting. We use part of the techniques developed in [Rug13] to prove the result.…”

Tropical Hopf manifolds and contracting germs

“…In this section we used the fact that K has characteristic 0. In positive characteristic, normal forms of contracting automorphisms f ∶ (A d , 0) → (A d , 0) coincide with the ones given in characteristic 0 (see [Vos91, Theorem 1.1], [Rug13,Theorem 2.7], [Rug15, Remark 7.2]). In dimension d = 2, the study of periodic curves of a germ f defined over a field of positive characteristic is similar to the situation in characteristic 0.…”