Abstract
Perez-Marco proved the existence of non-trivial totally invariantconnected compacts called hedgehogs near the fixed point of anonlinearizable germ of holomorphic diffeomorphism. We show thatif two nonlinearisable holomorphic germs with a common indifferentfixed point have a common hedgehog then they must commute. Thisallows us to establish a correspondence between hedgehogs andnonlinearizable maximal abelian subgroups of Diff($\mathbb{C},0$).We also show that two nonlinearizable germs with the same rotationnumber are conjugate if and only if a hedgehog of one can bemapped conformally onto a hedgehog of the other. Thus theconjugacy class of a nonlinearizable germ is completely determinedby its rotation number and the conformal class of its hedgehogs.
Highlights
We consider the dynamics of a holomorphic germ f (z) = e2πiαz + O(z2), α ∈ R − Q near the indifferent irrational fixed point 0
Cremer ([Cr1], [Cr2]) in the 1920’s showed the existence of nonlinearizable germs for rotation numbers very well approximable by rationals, while that of C.L.Siegel ([Si]) in 1942 and A.D.Brjuno in the 1960’s showed linearization was always possible for for germs with rotation numbers poorly approximated by rationals
Let K be a hedgehog for a nonlinearizable germ f
Summary
When the rotation number α is irrational, a Siegel compact which is not contained in the closure of a linearization domain of f is called a hedgehog. We restrict ourselves to the case of germs with irrational rotation number (so any Siegel compact is either a linearization domain, or a linearizable hedgehog, or a nonlinearizable hedgehog).
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