Surgery in Complex Dynamics. Carsten Lunde Petersen IMFUFA at NSM, Roskilde University Quasi-conformal massage, -maps on drugs -. p. 1/19
Introduction Magic Lecture 1: Cut and Past surgery using interpolations. Magic Lecture 2: Quasi-conformal massage, -maps on drugs -. Magic Lecture 3: Quasi-conformal surgery using real conjugacies. Magic Lecture 4: Trans-quasi-conformal surgery. Quasi-conformal massage, -maps on drugs -. p. 2/19
Wringing and stretching Wringing and stretching the complex structure Let s H + and define l s : C C by ls (z) = 1 2 (s + 1)z + 1 2 (s 1)z = (s 1)z x + z = sz x + iz y, where z = z x + iz y. Or written as an R-linear selfmap of R 2 : ls ( z x z y ) = ( s x 0 s y 1 ) ( Evidently l s is an o-p q-c homeo with constant dilatation z x z y µ s (z) = µ s = s 1 s + 1. Quasi-conformal massage, -maps on drugs -. p. 3/19 ).
Wringing and Stretching II Being an R-linear map l s (z) = sz x + iz y commutes with real multiplication and conjugates translation by Λ C to translation by l s (Λ): C e ls z rz C e ls C e ls z z+λ C e ls C z rz C. C z z+e l s (Λ) C. Furthermore l s (iy) = iy, and l s (x) = sx and thus ls (ir) = ir, ls (R) = sr and ls (R + iy) = sr + iy. Quasi-conformal massage, -maps on drugs -. p. 4/19
Wringing and Stretching III For s H + consider likewise the map l s : C C given by l s (z) = z z s 1 = z z z s An easy computation shows that C e z e ls C e z. C l s C. Quasi-conformal massage, -maps on drugs -. p. 5/19
Wringing and Stretching IV It immediately follows that l s : a) is an o-p q-c homeo with a coherent dilatation given by µ s (z) = z z s 1 s + 1 = log (z) log z µ s(log(z)). b) fixes pointwise the unit circle and maps the radial line exp(r + iy) to the logaritmic spiral exp(sr + iy), c) commutes with z z d for d N and conjugates multiplication by λ C to multiplication by l s (λ). d) for any z the map s µ s (z) is holomorphic (Möbius). Quasi-conformal massage, -maps on drugs -. p. 6/19
Wringing and Stretching V We define a Lie-group structure on H + by requirering that the map s l s is a contravariant group homomorphism, that is l s s = l s l s. An easy computation shows that this amounts to s s = s xs x + i(s ys x + s y ) = s x s + is y = l s (s) When stressing this group structure we shall write R for H +. Quasi-conformal massage, -maps on drugs -. p. 7/19
Wringing and Stretching VI The Lie-group (R, ) is non-commutative, but it is generated by the two one-dimensional commutative subgroups: Stretch: (S, ) = (R +, ) and Wring: (W, ) = (1 + ir, ). These two subgroups act on R from left by multiplication and by addition of the imaginary part repsectively: s = s x S, s R s s = ss, w = 1 + iw y W, s R w s = s + iw y. Quasi-conformal massage, -maps on drugs -. p. 8/19
Druging a map with an attracting orbit Theorem 1 (Douady and Hubbard). For any hyperbolic component H of the Mandelbrotset M the multiplier map λ : H D for the associated attracting periodic orbit is an isomorphism Theorem 2 (Eremenko and Lyubich). For any hyperbolic component H of the exponential family E κ (z) = exp(z + κ) the multiplier map λ : H D for the associated attracting periodic orbit is a universal covering map. Quasi-conformal massage, -maps on drugs -. p. 9/19
Druging II Common features: In both cases we can define a group action by (R, ) on the subset of H consisting of parameters for which the multiplier map is non-zero. In the quadratic case let H denote H deprived of its center(s). Applying this action to a fixed parameter in H defines a surjective and periodic map onto H. Passing to a quotient space essentially yields the result. In the exponential case applying this action to any parameter yields an injective and surjective map and hence the result. Quasi-conformal massage, -maps on drugs -. p. 10/19
The action Initially you may choose to think of f 1 as either a quadratic polynomial Q c or an exponetial map f κ. Suppose f1 k(z 1) = z 1 and (f1 k) (z 1 ) = λ 1 D. Let Λ 1 denote the immediate basin of z 1. We shall assume the singular (0)/ critical (c 1 ) value belongs to Λ 1. Let φ 1 : Λ 1 C denote the linearizer say normalized by φ 1 (0) = λ 1 (exponential) / φ 1 (c 1 ) = λ 1 (quadratic). For s R let σ s denote the unique f1 invariant almost complex structure, which satisfies: σ s = (l s φ 1 ) (σ 0 ) on Λ 1, σ s = σ 0 on C \ n 0 f n 1 (Λ 1 ) Quasi-conformal massage, -maps on drugs -. p. 11/19
The action II Let h s : C C be an integrating o-p q-c homeomorphism for σ s. As normalization conditions we take h s ( ) = and h s (0) = 0 in both cases, however: If f 1 (z) = exp(z + κ 1 ) we demand h s (i2π) = i2π. If f 1 (z) = z 2 + c 1 we demand h s (z)/z 1 as z. In the later case we find easily that f s (z) := h s f 1 h 1 s = z 2 + c s, where c s = h s (c 1 ). Quasi-conformal massage, -maps on drugs -. p. 12/19
The action III In the first exponential case we need first to realize that σ s satisfies σ s (z + i2π) = σ s (z), in order to conclude that h s (z + i2π) = h s (z) + i2π. Then it follows easily that f s (z) := h s f 1 h 1 s = h s (e κ 1 ) e z. Taking κ s = log(h s (e κ 1 )) to be the unique continuous and thus holomorphic choice of logarithm sending 1 to κ 1, we have f s (z) = exp(z + κ s ). Quasi-conformal massage, -maps on drugs -. p. 13/19
The action IV We have holomorphic mappings into parameter spaces s κ s, c s : H + C such that f k s (z s ) = z s := h s (z 1 ), and (f k s ) (z s ) = λ s = l s (λ). Where either f s (z) = exp(z + κ s ) or f s (z) = z 2 + c s. These mappings into parameterspaces are in fact group actions on H : s, s R : κ s s = (κ s ) s, and c s s = (c s ) s. Quasi-conformal massage, -maps on drugs -. p. 14/19
Exploiting the group action Notice first that by the group action if some map g 1 = f s for some s R, then g s 1 = f 1 and the two group action image sets are equal: {f s s R} = {g s s R}. Moreover this set is a neighbourhood of f 1 (and of g 1 ). Since H is connected it follows that H = {f s s R} Secondly by the group action and the above it suffices to consider the case λ 1 = e 1 so that λ s = l s (λ 1 ) = e s. Quasi-conformal massage, -maps on drugs -. p. 15/19
Exploiting the group action II Thirdly by the group action property if f s = f s for some s s R then f s s 1 = f 1 and hence s 1 = s s 1 = 1 + n2πi for some n Z as e 1 = λ 1 = λ s0 = e s 0. Moreover f s = f s1 s for all s R. Suppose s 1 = 1 + n2πi satisfies f s = f s1 s for all s R. Then s k = s k 1 = 1 + kn2πi also satisfies f s = f sk s for all s R. Thus we can suppose s 1 = 1 + n2πi with n N minimal. Quasi-conformal massage, -maps on drugs -. p. 16/19
Exploiting the group action II Then the surjective, holomorphic and n2πi periodic map s f s from H + to H descends to a biholomorphic map F = w f w : Dstar H given by f w = f n log w and the multiplier map λ on H is the n-fold covering (F 1 ) n of D. In the case of quadratic polynomials one can prove that the above holds with n = 1 so that the multiplier map is an isomorphism and extends to a biholomorphic map of H onto D. In the case of the exponential family, which we shall pursue here we need to prove that there is no n such that f s1 = f 1 with s 1 = 1 + n2πi: Quasi-conformal massage, -maps on drugs -. p. 17/19
Proof of Eremenko-Lyubich Th. Suppose to the contrary that for some n N f s1 = f 1 with s 1 = 1 + n2πi. Then by the above there is a biholomorphic map K : D H such that E K(w) = exp(z + K(w)) = f n log w has a k-periodic orbit of mulitplier w n. That is H is a punctured disk with puncture κ 0 C. Let us first notice that for any κ > 0, the singular value 0 of E κ iterates to so that R + H =. Thus κ 0. Quasi-conformal massage, -maps on drugs -. p. 18/19
Proof of Eremenko-Lyubich Th. II Let z 0 (κ),... z k 1 (κ) denote the attracting k-periodic orbit of E κ, κ H. As κ approaches κ 0 the multiplier of this orbit converge to 0 by the above. Hence some point say z 0 (κ) converges to 0 and R(z k 1 (κ)). However by continuity z k 1 (κ) = Eκ k 1 (z 0 (κ)) Eκ k 1 0 (0) C, as κ κ 0. Thus s f s is injective in the exponential case. That is s κ(s) : H + H is biholomorphic and the multiplier map λ(κ) = e s(κ) is a universal covering. Quasi-conformal massage, -maps on drugs -. p. 19/19