Bifurcation currents in one-dimensional holomorphic dynamics

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4 Equidistribution towards the bifurcation current 514.1 Distribution of critically periodic parameters . . . . . . . . . . . . . . 514.1.1 Statement and a general strategy . . . . . . . . . . . . . . . . 514.1.2 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . 544.2 Distribution of rational maps with cycles of a given multiplier . . . . 564.2.1 The case of attracting cycles . . . . . . . . . . . . . . . . . . . 564.2.2 Averaging the multipliers . . . . . . . . . . . . . . . . . . . . . 594.2.3 The case of neutral cycles in polynomial families . . . . . . . . 634.3 Laminated structures in bifurcation loci . . . . . . . . . . . . . . . . . 654.3.1 Holomorphic motion of the Mandelbrot set . . . . . . . . . . . 654.3.2 Further laminarity statements for T bif . . . . . . . . . . . . . . 695 The bifurcation measure 715.1 A Monge-Ampère mass related with strong bifurcations . . . . . . . . 715.1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.2 Some concrete families . . . . . . . . . . . . . . . . . . . . . . 735.2 Density statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.1 Misiurewicz parameters . . . . . . . . . . . . . . . . . . . . . . 755.2.2 Shishikura or hyperbolic parameters . . . . . . . . . . . . . . 775.2.3 Shishikura parameters with chosen multipliers . . . . . . . . . 795.3 The support of the bifurcation measure . . . . . . . . . . . . . . . . . 795.3.1 A transversality result . . . . . . . . . . . . . . . . . . . . . . 805.3.2 The bifurcation measure and strong-bifurcation loci . . . . . . 815.3.3 Hausdorff dimension estimates . . . . . . . . . . . . . . . . . . 846
Chapter 1Rational functions as ergodicdynamical systems1.1 Potential theoretic aspects1.1.1 The Fatou-Julia dichotomyA rational function f is a holomorphic map of the Riemann sphere to itself and maybe repesented as the ratio of two polynomialsf = a 0+a 1 z+a 2 z 2 +···+a d z db 0 +b 1 z+b 2 z 2 +···+b d z dwhere at least one of the coefficents a d and b d is not zero. The number d is thealgebraic degree of f. In the sequel we shall more likely speak of rational map. Sucha map may also be considered as a holomorphic ramified self-cover of the Riemannsphere whose topological degree is equal to d. Among these maps, polynomials areexactly those for which ∞ is totally invariant: f −1 {∞} = f{∞} = ∞.It may also be convenient to identify the Riemann sphere with the one-dimensionalcomplex projective space P 1 that is the quotient of C 2 \ {0} by the action z ↦→ u · zof C ∗ . Let us recall that the Fubini-Study form ω on P 1 satisfies π ∗ (ω) = dd c ln ‖ ‖where the norm is the euclidean one on C 2 .In this setting, the map f can be seen as induced on P 1 by a non-degenerate andd-homogenous map of C 2F (z 1 , z 2 ) := ( a 0 z d 2 + a 1 z 1 z d−12 + · · · + a d z d 1, b 0 z d 2 + b 1 z 1 z d−12 + · · · + b d z d 1through the canonical projection π : C 2 \ {0} → P 1The homogeneous map F is called a lift of f; all other lifts are proportional to F .A point around which a rational map f does not induce a local biholomorphismis called critical. The image of such a point is called a critical value. A degreed rational map has exactly (2d − 2) critical points counted with multiplicity. The7)
Page 3: IntroductionThe aim of these lectur
Page 8 and 9: critical set of f is the collection
Page 10 and 11: Lemma 1.1.6 Let F be a lift of a de
Page 12 and 13: and is a global potential of the Gr
Page 14: 1.2.2 The natural extensionTo any e
Page 18 and 19: 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
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dd c L r n = ∫ d−n 2π[P er 2π
Page 63 and 64:
Remark 4.2.8 Using standard techniq
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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 and 70:
Theorem 4.3.5 Let σ : (( Ω hyp \
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Chapter 5The bifurcation measure5.1
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h = u on the boundary of B rdd c h
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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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