Discrete Holomorphic Local Dynamical Systems

Discrete Holomorphic Local Dynamical Systems 31
Then Theorem 4.10 is consequence of what we have seen and the following
Theorem 4.19 (Yoccoz, 1988 [Y2]).
(i) B(θ) < +∞ if and only if λ = e 2πiθ satisfies Brjuno’s condition (27);
(ii) there exists a universal constant C > 0 such that
|logr(θ)+B(θ)|≤C
for all θ ∈ [0,1) \ Q such that B(θ) < +∞;
(iii) if B(θ) =+∞ then there exists a non-linearizable f ∈ End(C,0) with
multiplier e 2πiθ .
If 0 is a Siegel point for f ∈ End(C,0), the local dynamics of f is completely
clear, and simple enough. On the other hand, if 0 is a Cremer point of f ,then
the local dynamics of f is very complicated and not yet completely understood.
Pérez-Marco (in [P3, 5–7]) and Biswas ([B1, 2]) have studied the topology and the
dynamics of the stable set in this case. Some of their results are summarized in the
following
Theorem 4.20 (Pérez-Marco, 1995 [P6, 7]). Assume that 0 is a Cremer point for
an elliptic holomorphic local dynamical system f ∈ End(C,0). Then:
(i) ThestablesetKfis compact, connected, full (i.e., C\Kf is connected), it is not
reduced to {0}, and it is not locally connected at any point distinct from the
origin.
(ii) Any point of Kf \{0} is recurrent (that is, a limit point of its orbit).
(iii) There is an orbit in Kf which accumulates at the origin, but no non-trivial orbit
converges to the origin.
Theorem 4.21 (Biswas, 2007 [B2]). The rotation number and the conformal class
of Kf are a complete set of holomorphic invariants for Cremer points. In other
words, two elliptic non-linearizable holomorphic local dynamical systems f and g
are holomorphically locally conjugated if and only if they have the same rotation
number and there is a biholomorphism of a neighbourhood of Kf with a
neighbourhood of Kg.
Remark 4.22. So, if λ ∈ S 1 is not a root of unity and does not satisfy Brjuno’s condition
(27), we can find f1, f2 ∈ End(C,0) with multiplier λ such that f1 is holomorphically
linearizable while f2 is not. Then f1 and f2 are formally conjugated
without being neither holomorphically nor topologically locally conjugated.
Remark 4.23. Yoccoz [Y2] has proved that if λ ∈ S 1 is not a root of unity and does
not satisfy Brjuno’s condition (27) then there is an uncountable family of germs
in End(C,O) with multiplier λ which are not holomorphically conjugated to each
other nor holomorphically conjugated to any entire function.
See also [P2, 4] for other results on the dynamics about a Cremer point. We
end this section recalling the somewhat surprising fact that in the elliptic case
the multiplier is a topological invariant (in the parabolic case this follows from
Proposition 3.20):