Discrete Holomorphic Local Dynamical Systems

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Dynamics of Entire Functions 305 (Sometimes, one also uses the lower order of f ; for this definition, the limsup is replaced by the liminf.) Remark 1.16. The prototypical example of a function of finite order d ∈ N is z ↦→ exp(z d ). Unlike the properties of finite or bounded type, the property of finite order is not preserved under compositions, and was thus thought to be of limited use in dynamics; however, the order of growth has recently found important applications in dynamics, for instance when dynamical properties are controlled using geometric function theory (for instance, in Theorem 4.11 below). The finite order condition also comes into play in the discussion of direct and indirect asymptotic values. Theorem 1.17 (Denjoy-Carleman-Ahlfors). For any function f of finite order, the number of asymptotic values, and even the number of different asymptotic tracts, is at most max{1,2ordf } and hence finite. More precisely, if f has m direct and n indirect asymptotic tracts, then 2m + n ≤ max{1,2ordf }. In fact, between any two different asymptotic tracts of finite asymptotic values, there must be an asymptotic tract of the asymptotic value ∞; and if the latter is very narrow, then f must grow very quickly along this tract; the extra count for direct asymptotic tracts comes in because these, too, need to have certain width (postcomposing f with a Möbius transform preserves the order, and can turn direct tracts over finite asymptotic values to tracts over ∞). Compare [Ne53, Sec. 269]. Theorem 1.18 (Indirect Asymptotic Values and Critical Values). For any function of finite order, every indirect asymptotic value is a limit point of critical values. For a proof, see [BE95]. The following result from [Z75] is often useful in complex dynamics, especially transcendental dynamics. Lemma 1.19 (The Zalcman Lemma). Let U ⊂ C be any domain and let F be a non-normal family of holomorphic functions from U to C. Then there exist a non-constant entire function g: C → C and sequences fk ∈ F ,zk ∈ U and ρk > 0 with ρk → 0 so that zk → z∞ ∈ U and gk(ζ) := fk(zk + ρkζ) −→ g(ζ) uniformly on compacts in C.Ifa∈ U is such that the restriction of F to any neighborhood of a is not normal, then we may require that zn → a. Proof. After rescaling, we may suppose that D ⊂ U and that the point a = 0has the property that the restriction of F to any neighborhood of 0 is not normal. Let ( fk) ⊂ F be any sequence that does not form a normal family when restricted to any neighborhood of a.
306 Dierk Schleicher According to Marty’s criterion (Theorem A.3), a family of maps ( fk) is normal if and only if its spherical derivatives f ♯ (z)=| f ′ (z)|/(1+| f (z)| 2 ) are locally bounded. Since this is not the case, one can extract a subsequence of the fk and then a sequence ζk → 0 with f ♯ k (ζk) → ∞. Choose asequence (rk) ⊂ R + so that rk → 0butrkf ♯ k (ζk) → ∞; we may assume that |ζk| + 2rk < 1forallk. Find a sequence zk ∈ C with |zk − ζk| < rk so that Mk := f ♯ k (zk)(1 −|zk − ζk|/rk)= sup |z−ζ k|
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Lecture Notes in Mathematics 1998 E
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Marco Abate · Eric Bedford · Marc
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Preface The theory of holomorphic d
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Preface vii here are of independent
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x Contents 2.7 Cremona Representati
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List of Contributors Marco Abate Di
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2 Marco Abate on U ∩ f −1 (U) o
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4 Marco Abate without constant term
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6 Marco Abate Proof. Let us assume
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8 Marco Abate Definition 2.8. The n
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10 Marco Abate Then it is easy to c
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12 Marco Abate When |w| is so large
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14 Marco Abate As a consequence, th
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16 Marco Abate where β is a formal
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18 Marco Abate a lower half-plane.
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20 Marco Abate Definition 3.19. The
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22 Marco Abate Remark 4.5. The same
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24 Marco Abate Remark 4.11. Conditi
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26 Marco Abate defined for |t| smal
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28 Marco Abate � � � card 0
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30 Marco Abate to show (see, e.g.,
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32 Marco Abate Theorem 4.24 (Naishu
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34 Marco Abate it escapes both in f
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36 Marco Abate as I know, the probl
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38 Marco Abate 6 Several Complex Va
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40 Marco Abate Remark 6.8. There ar
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42 Marco Abate (ii) f |M is holomor
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44 Marco Abate In [ABT1] we proved
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46 Marco Abate and dimE = 1; but th
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48 Marco Abate It turns out that σ
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50 Marco Abate and the eigenvalues
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52 Marco Abate [CD] Coman, D., Dabi
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54 Marco Abate [Ni] Nishimura, Y.:
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Dynamics of Rational Surface Automo
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166 Tien-Cuong Dinh and Nessim Sibo
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Page 263 and 264: 254 Tien-Cuong Dinh and Nessim Sibo
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Page 307 and 308: 298 Dierk Schleicher In a brief app
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Page 339 and 340: 330 Dierk Schleicher Definition 7.1
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Discrete Holomorphic Local Dynamical Systems

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