Bifurcation currents in one-dimensional holomorphic dynamics

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The proof is similar to that of Theorem 4.3.6 but requires a special treatmentfor the extension problems since there is no λ-lemma available. The key is to usethe fact that, by construction, the starting motion σ : (∪ m≠n (U m ∩ U n )) × ∆ −→ U nsatisfies the following property:p n (σ(λ, t), t) = 0; ∀λ ∈ Bifn c , ∀t ∈ ∆.Such a motion is what we call a p n -guided holomorphic motion. Using Zalcmanrescaling lemma, one proves the following compactness property for guided holomorphicmotions. We stress that here, an holomorphic motion G is seen as familyof disjoints holomorphic discs σ and G t0 is the set of points σ(t 0 ).Theorem 4.3.7 Let p(λ, w) be a polynomial on C 2 × C such that the degree ofp(·, w) does not depend on w ∈ ∆. Let G be a p-guided holomorphic motion in C 2such that any component of the algebraic curve {p(·, t) = 0} contains at least threepoints of G t for every t ∈ ∆. Then, for any F ⊂ G such that F t0 is relatively compactin C 2 for some t 0 ∈ ∆, there exists a continuous p-guided holomorphic motion ̂F inC 2 such that F ⊂ ̂F and ̂F t0 = F t0 .The above Theorem plays the role of the λ-lemma in the proof of Theorem 4.3.6.We refer to the paper [BB2] for details.We will now end this section by quoting another laminarity result for T bif whichis due to Dujardin (see [Du])Theorem 4.3.8 Within the polynomial family of degree 3, the bifurcation currentis laminar outside the connectedness locus.We refer to section 2.3 for the discussion of the connectedness locus within polynomialfamilies.70
Chapter 5The bifurcation measure5.1 A Monge-Ampère mass related with strongbifurcations5.1.1 Basic propertiesSince the bifurcation current T bif of any holomorphic family (f λ ) λ∈Mof degree drational maps has a continuous potential L (see Definition 3.2.6 and Theorem 3.2.9),one may define the powers (T bif ) k := T bif ∧ T bif ∧ · · · ∧ T bif for any k ≤ m := dimM.We recall that for any closed positive current T , the product dd c L ∧ T is definedby dd c L ∧ T := dd c (LT ). In particular, (T bif ) m is a positive measure on M which isequal to the Monge-Ampère mass of the Lyapunov function L.Definition 5.1.1 Let (f λ ) λ∈Mbe a holomorphic family of degree d rational mapsparametrized by a complex manifold M of dimension m. The bifurcation measureµ bif of the family is the positive measure on M defined byµ bif = 1 m! (T bif) m = 1 m! (ddc L) mwhere T bif is the bifurcation current and L the Lyapunov function of the family.The following proposition is a direct consequence of the definition and the factthat µ bif has locally bounded potentials.Proposition 5.1.2 The support of µ bifµ bif does not charge pluripolar sets.is contained in the bifurcation locus andIt is actually possible to define the bifurcation measure in the moduli space Mod dof degree d rational maps and show that this measure has strictly positive and finitemass (see [BB1] Proposition 6.6). Although all the results we will present here aretrue in Mod d , we will restrict ourself to the technically simpler situation of holomorphicfamilies. The example we have in mind are the polynomial families and71
Page 3:
IntroductionThe aim of these lectur
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4 Equidistribution towards the bifu
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critical set of f is the collection
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Lemma 1.1.6 Let F be a lift of a de
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and is a global potential of the Gr
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1.2.2 The natural extensionTo any e
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Theorem 1.3.5 The Lyapunov exponent
Page 20 and 21: We have to show that µ L n → µ
Page 22 and 23: This can be easily seen : if û ∈
Page 25 and 26: Thus, when considering degree d pol
Page 27 and 28: 1- for any w ∈ C \ {1}, the funct
Page 29 and 30: Let us now consider the setsΛ ∗
Page 31 and 32: We finally consider two subcases.(
Page 33 and 34: iv) P is conformally conjugated to
Page 35 and 36: 1) G(c, a) ≤ ln max{|a|, |c k |}
Page 37 and 38: Chapter 3The bifurcation currentWe
Page 39 and 40: Let us mention here that there exis
Page 41 and 42: Proof. The equivalence between 1) a
Page 43 and 44: Definition 3.1.16 The set M is of d
Page 45 and 46: Remark 3.2.2 As the Green function
Page 47 and 48: Remark 3.2.5 In [BB1] the formula 3
Page 49 and 50: Proof. According to Theorem 3.2.4 a
Page 51 and 52: Chapter 4Equidistribution towards t
Page 53 and 54: We will also need to following clas
Page 55 and 56: (h − g)(λ 0 ) < 0. As the functi
Page 57 and 58: Since, by definition, T bif = dd c
Page 59 and 60: ɛ n (λ, w) = d −n∑1≤|w n,j
Page 61 and 62: dd c L r n = ∫ d−n 2π[P er 2π
Page 63 and 64: Remark 4.2.8 Using standard techniq
Page 65 and 66: lim k d −k [Per k (0)] = T bif ),
Page 67 and 68: Let us also set U n,1 := U n ∩ U
Page 69: Theorem 4.3.5 Let σ : (( Ω hyp \
Page 73 and 74: h = u on the boundary of B rdd c h
Page 75 and 76: counted with multiplicity near 0 fo
Page 77 and 78: 1limn j →∞ d [P reper n nj j] +
Page 79 and 80: 5.2.3 Shishikura parameters with ch
Page 81 and 82: Theorem 5.3.3 The activity map χ n
Page 83 and 84: using the homogeneity property of t
Page 85 and 86: Bibliography[A] Magnus Aspenberg. P
Page 87: Roland Roeder, Sylvain Bonnot, Davi
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Bifurcation currents in one-dimensional holomorphic dynamics

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