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Holomorphic Dynamics and Hyperbolic Geometry Shaun Bullett ...

1 Introduction1.1 OverviewThe objective of these lecture notes is to develop some of the main themes in the study of iterated rationalmaps, that is to say maps of the Riemann sphere Ĉ = C ∪ ∞ to itself of the formz → p(z)q(z)(where p **and** q are polynomials with complex coefficients), **and** the study of Kleinian groups, discrete groups ofmaps from Ĉ to itself, each of the formz → az + bcz + d(where a, b, c, d are complex numbers, with ad − bc ≠ 0).We shall develop these studies in parallel: although there is no single unifying theory encompassing both areasthere are tantalizing similarities between them, **and** results in one field frequently suggest what we should lookfor in the other (Sullivan’s Dictionary).The study of iterated rational maps had its first great flowering with the the work of the French mathematiciansJulia **and** Fatou around 1918-20, though its origins perhaps lie earlier, in the late 19th century, in the moregeometric work of Schottky, Poincaré, Fricke **and** Klein. It has had its second great flowering over the last 30years, motivated partly by the spectacular computer pictures which started to appear from about 1980 onwards,partly by the explosive growth in the subject of chaotic dynamics which started about the same time, **and** notleast by the revolutionary work in three-dimensional hyperbolic geometry initiated by Thurston in the early1980’s. In the intervening period Siegel (in the 1940s) had proved key results concerning local linearisabilityof holomorphic maps, **and** Ahlfors **and** Bers (in the 1960s) following pioneering work of Teichmüller (in thelate 1930s) had developed quasiconformal deformation theory for Kleinian groups: the stage was set for anexplosion of interest, both experimental **and** analytical. Some of the names associated with this second greatwave of activity are M**and**elbrot, Douady, Hubbard, Sullivan, Herman, Milnor, Thurston, Yoccoz, McMullen**and** Lyubich. Both subjects are still very active indeed: as we shall see, some of the major conjectures arestill waiting to be proved. But the remarkable mixture of complex analysis, hyperbolic geometry **and** symbolicdynamics that constitutes the subject of holomorphic dynamics yields powerful methods for problems whichat first sight might appear only to concern only real mathematics. For example the most conceptual proof ofthe universality of the Feigenbaum ratios for period doubling renormalisation of real unimodal maps is that ofSullivan (1992) using complex analysis.We start our study of rational maps **and** Kleinian groups - as we mean to go on - with motivating examples.1.2 The family of maps z → z 2 + c(i) c = 0Here the dynamical behaviour is straightforward. When we iterate z → z 2 any orbit started inside the unitcircle heads towards the point 0, any orbit started outside the unit circle heads towards ∞, **and** any orbit startedon the unit circle remains there. The two components of {z : |z| ≠ 1} are known as the Fatou set of the map**and** the circle |z| = 1 is called the Julia set. On the unit circle itself the dynamics are those of the shift, namelyif we parametrise the circle by t ∈ [0, 1) ⊂ R (t = arg(z)/2π): then z → z 2 sends t → 2t mod 1.Any t ∈ [0, 1) of the form t = m/(2 n − 1) (for 0 ≤ m < 2 n − 1 integer) is periodic, of period n (exercise: provethis). Hence the periodic points form a dense set on the unit circle. Moreover the map z → z 2 has sensitivedependence on initial condition, since the map on the unit circle doubles distance.2

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- Page 5 and 6: 2 Dynamics of rational maps2.1 The
- Page 7 and 8: For the converse, let f : Ĉ → Ĉ
- Page 9 and 10: Proof Any ‘circle’ in Ĉ (inclu
- Page 11 and 12: 2.6 Conjugacies, fixed points and m
- Page 13 and 14: 3 Fatou and Julia setsThe following
- Page 15 and 16: compare it with Picard’s Theorem
- Page 18 and 19: Figure 6: The circle γ 0 and its i
- Page 20 and 21: f(U) ⊂ U and f| U is injective. L
- Page 22 and 23: Case 2: λ = e 2πiα with α irrat
- Page 24 and 25: (where r 2 = x 2 1 + x 2 2 + x 2 3)
- Page 26 and 27: Definition The set of all z ∈ Ĉ
- Page 28 and 29: in Ĉ − Λ(G)). The set C(G) is c
- Page 30 and 31: 6 Fundamental domains and examples
- Page 32 and 33: Definition A side-pairing transform
- Page 34 and 35: Figure 12: The limit set of a trunc
- Page 36 and 37: Figure 13: The Mandelbrot setφ c (
- Page 38 and 39: Proof The points of period 1 or 2 a
- Page 40 and 41: Figure 16: θ − (2/5) = .01001 =
- Page 42 and 43: 8 Quasiconformal mappings: the Meas
- Page 44 and 45: 8.3 1st Application: maps in the sa
- Page 46 and 47: Figure 21: The Straightening Theore
- Page 48 and 49: References[1] L.Ahlfors, Complex An
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Holomorphic Dynamics and Hyperbolic