An Integral Equality and its Applications by Hille E.

By Hille E.

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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]).

Then the following statements are equivalent: (i) the origin is a Siegel point for the quadratic polynomial fλ (z) = λ z + z2 ; (ii) the origin is a Siegel point for all f ∈ End(C, 0) with multiplier λ ; (iii) the number λ satisfies Brjuno’s condition +∞ 1 1 ∑ 2k log Ωλ (2k+1) < +∞. 7 (see also [He] and references therein) has proved that condition (iii) implies condition (ii). Yoccoz, using a more geometric approach based on conformal and quasi-conformal geometry, has proved that (i) is equivalent to (ii), and that (ii) implies (iii), that is that if λ does not satisfy (27) then the origin is a Cremer point for some f ∈ End(C, 0) with multiplier λ — and hence it is a Cremer point for the quadratic polynomial fλ (z).

Let λ ∈ S1 be such that lim sup m→+∞ 1 1 log = +∞. m Ωλ (m) (24) Then there exists f ∈ End(C, 0) with multiplier λ which is not holomorphically linearizable. Furthermore, the set of λ ∈ S1 satisfying (24) contains a Gδ -dense set. Proof. Choose inductively a j ∈ {0, 1} so that |a j + X j | ≥ 1/2 for all j ≥ 2, where X j is as in (23). Then f (z) = λ z + a2z2 + · · · ∈ C0 {z}, while (24) implies that the radius of convergence of the formal linearization h is 0, and thus f cannot be holomorphically linearizable, as required.

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