By Nessim Sibony, Dierk Schleicher, Dinh Tien Cuong, Marco Brunella, Eric Bedford, Marco Abate, Graziano Gentili, Giorgio Patrizio, Jacques Guenot

The idea of holomorphic dynamical structures is a topic of accelerating curiosity in arithmetic, either for its hard difficulties and for its connections with different branches of natural and utilized arithmetic. This quantity collects the Lectures held on the 2008 CIME consultation on "Holomorphic Dynamical platforms" held in Cetraro, Italy. This CIME path enthusiastic about a few vital issues within the learn of discrete and non-stop dynamical platforms, together with either neighborhood and worldwide points, offering a desirable advent to many key difficulties in present examine. The contributions offer an considerable description of the phenomena happening in critical topics of holomorphic dynamics equivalent to automorphisms and meromorphic self-maps of projective areas, of complete maps on advanced areas and holomorphic foliations in surfaces and better dimensional manifolds, elaborating at the various strategies used and familiarizing readers with the newest findings on present learn topics.

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**Example text**

For all m ≥ 2 we have Nm (k) ≤ 0, 2k m if k ≤ m , − 1, if k > m. Proof. We argue by induction on k. If l ≤ k ≤ m we have εl ≥ Ωλ (m), and hence Nm (k) = 0. Assume now k > m, so that 2k/m − 1 ≥ 1. Write δk as in (31); we have a few cases to consider. Case 1: εk ≥ 14 Ωλ (m). Then N(k) = N(k1 ) + · · · + N(kν ), and applying the induction hypothesis to each term we get N(k) ≤ (2k/m) − 1. Case 2: εk < 14 Ωλ (m). Then N(k) = 1 + N(k1) + · · · + N(kν ), and there are three subcases. 1: k1 ≤ m. Then N(k) = 1 ≤ 2k − 1, m and we are done.

Then there exist: (a) a 2-dimensional complex manifold M (obtained by blowing-up a finite number of points; see next section); (b) a surjective holomorphic map π : M → C2 such that the restriction π |M\E : M \ E → C2 \ {O} is a biholomorphism, where E = π −1 (O); (c) a point p ∈ E; and (d) a rigid holomorphic germ f˜ ∈ End(M, p) so that π ◦ f˜ = f ◦ π . Discrete Holomorphic Local Dynamical Systems 35 See also Ruggiero [Ru] for similar results for semi-superattracting (one eigenvalue zero, one eigenvalue different from zero) germs in C2 .

Thus if there are no resonances then f is smoothly linearizable. Even without resonances, the holomorphic linearizability is not guaranteed. 15 (Poincar´e, 1893 [Po]). Let f ∈ End(Cn , O) be a locally invertible holomorphic local dynamical system in the Poincar´e domain. Then f is holomorphically linearizable if and only if it is formally linearizable. In particular, if there are no resonances then f is holomorphically linearizable. Reich [Re2] describes holomorphic normal forms when d fO belongs to the Poincar´e ´ domain and there are resonances (see also [EV]); P´erez-Marco [P8] discusses the problem of holomorphic linearization in the presence of resonances (see also Raissy [R1]).