Spherical Universality of Composition Operators and Applications in Rational Dynamics

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1 Spherical Universality of Composition Operators and Applications in Rational Dynamics Andreas Jung and Jürgen Müller March 9, 2016 Abstract Let D be an open subset of the complex plane and let f be a holomorphic self-map of D. We deduce sufficient conditions on D and on f such that there exists some meromorphic function g for which the orbit {g f n : n N} is dense in M (D). Using the well-known geometric behavior of certain finite Blaschke products on the unit disk, these results will be applied to a particular situation which occurs in the theory of complex dynamics, namely to the case that f is a rational function which has an attracting fixed point. Key words: universality, composition operator, finite Blaschke products MSC2010: 30K20, 37F10, 30J10 1 Introduction Given topological spaces X, Y and a family {T ι : ι I} of continuous functions T ι : X Y, an element x X is called universal for {T ι : ι I} if the set {T ι (x) : ι I} is dense in Y. In case that T : X X is continuous and T n := T... T denotes the n-th iterate of T, an element x X is called universal for T if it is universal for the family {T n : n N}, i.e. if its orbit {T n x : n N} is dense in X. We say that a property is fulfilled by comeager many elements of a Baire space X if it is fulfilled on a comeager subset of the corresponding space, i.e. this set contains a dense G δ -set in X. According to the Birkhoff transitivity theorem (see e.g. [10, Theorem 1.16]), in the case of a Polish space X a universal element for T exists if and only if T is topologically transitive on X and in this case comeager many elements turn out to be universal. An intensively investigated class of operators consists in the class of composition operators on spaces of holomorphic functions on open subsets of the complex plane; see e.g. [1] and [10]. For an open subset D of the complex plane C and a The first author has been supported by the Stipendienstiftung Rheinland-Pfalz. 1

2 holomorphic self-map f of D, the composition operator with symbol f is defined by C f : H(D) H(D), C f (g) := g f where H(D) denotes the Fréchet space of functions holomorphic in D endowed with the topology of locally uniform convergence. In the sequel we write M := M \ {0} for M C. For fixed 0 < λ < 1 let f denote the automorphism f : C C given by f(z) = λz for z C. In [4] (see also [9]) it was proved that the set of functions g H(C ) which have the property that (C f ) n g = g f n is dense in H(U) for all simply connected open subsets U of C is a comeager set in H(C ). The necessity of restricting to simply connected open subsets U of C can easily be seen by an application of the maximum modulus principle (see [4, Remark p.55]). In particular, the operator C f is not topologically transitive on H(C ) (see also [10, Corollary 4.30, Proposition 4.31]). One aim of this work is to show that the situation changes in an essential way if we consider compositional universality in the spherical metric. This will be done in Section 2, where firstly a general result on compositional universality in the spherical setting will be proved. Subsequently, this result will be applied to the case of holomorphic symbols having an attracting fixed point. Finally, using known results on the geometric behavior of finite Blaschke products of degree two on the unit disk which have an attracting fixed point at the origin, in Section 3 a geometric approach of obtaining compositional universality in case of rational symbols of degree two is worked out. 2 Compositional Universality in M (D) For an open set D C, we say that a function f which maps D to the extended complex plane C is spherically meromorphic if each restriction of f to a connected component of D is either meromorphic or else constantly infinity. We write M (D) for the set of spherically meromorphic functions on D. Then M (D) endowed with the topology of spherically locally uniform convergence turns out to be a completely metrizable space (cf. [6, Chapter VII]). A metric on M (D) inducing its topology is given by ρ(f, g) := sup min ( 1/n, ρ n (f, g) ), n N where ρ n (f, g) := max z Kn χ(f(z), g(z)) with χ denoting the chordal distance on C and (K n ) being a compact exhaustion of D. A basis of the topology of spherically locally uniform convergence is given by {V ε,k,d (g) : ε > 0, K D compact, g M (D)}, where V ε,k,d (g) := { h M (D) : max z K χ(h(z), g(z)) < ε }. 2

3 For a holomorphic self-map f of D we consider the composition operator C f : M (D) M (D), C f (g) := g f. Then C f is continuous with (C f ) n = C f n for all n N. Studying the proofs of the main results in [4] and [9], one can recognize the following rough outline of how to prove universality properties of a composition operator: It is assumed (and even necessary) that the symbol is injective and that the sequence of iterates of the symbol shows some kind of run-away behavior (cf. Definition 2.1 below). Subsequently, a suitable function is constructed in such a way that it can be approximated uniformly by rational functions having poles only outside a given compact set. This approximation, which is the crucial step of the proofs in [4] and [9], is obtained by an application of Runge s theorem on rational approximation of holomorphic functions. In order to obtain universality in the meromorphic setting, we need the following variant of Runge s theorem (see e.g. [8, Satz 11.1]) which is a straightforward application of the classical version: Let K C be compact, U K open, g M(U) and ε > 0. Then there exists a rational function R such that g R is holomorphic on an open neighborhood of K and such that max g(z) R(z) < ε. z K As an immediate consequence we obtain that for arbitrary open sets D in C the rational functions (restricted to D) are dense in M (D). We refer to this fact as spherical Runge theorem. In particular, the spherical Runge theorem implies that M (D) is separable and thus a Polish space. We now fix an open set D C, an open set Ω C with Ω D and a holomorphic function f : D D. Definition 2.1. We denote by U(D, Ω, f) the set of all open subsets U of D for which we have locally uniform convergence f n U Ω (i.e. the sequence (z dist(f n (z), Ω)) n N converges to 0 locally uniformly on U) and injectivity of each iterate f n on U. For U D open, a function g M (Ω) is called M (U)-universal for C f if the set {g f n U : n N} is dense in M (U), that is, g is universal for the sequence (C f n,u ) of composition operators C f n,u : M (Ω) M (U), C f n,u (g) := g f n U. Similarly to the holomorphic setting (cf. [11, Theorem 2.2]), the following universality result holds: Theorem 2.2. Let U U(D, Ω, f). Then comeager many functions in M (Ω) are M (U)-universal for C f. 3

4 Proof: According to the universality criterion (see e.g. [10, Theorem 1.57]), it suffices to show that the sequence (C f n,u ) is topologically transitive. In order to do so, let g M (Ω), h M (U) as well as K Ω, L U compact and ε > 0 be given. Because of U U(D, Ω, f), we have uniform convergence f n L Ω. Thus, setting δ := dist(k, Ω), there exists an N N with dist(f N (z), Ω) < δ for all z L, which implies K f N (L) =. Moreover, due to U U(D, Ω, f), the restriction f N U is injective so that the function { g(z), if z K ϕ : K f N (L) C, ϕ(z) := h ( (f N U ) 1 (z) ), if z f N (L) is well-defined. As the disjoint sets K and f N (L) are compact with K Ω and f N (L) f N (U) and since we have g M (Ω) as well as h M (U), we see that ϕ can be extended spherically meromorphic to an open neighborhood of K f N (L). Hence, the spherical Runge theorem yields a rational function R such that max χ(ϕ(z), R(z)) < ε. z K f n (L) In particular, due to ϕ = g on K we obtain max z K χ(g(z), R(z)) = max z K χ(ϕ(z), R(z)) max χ(ϕ(z), R(z)) < ε z K f N (L) and thus R Ω V ε,k,ω (g). Because of ϕ = h (f N U ) 1 on f N (L), we further obtain χ ( h(z), R(f N (z)) ) max z L = max w f N (L) χ ( h(z), C f N,U (R Ω ) (z) ) = max z L χ ( h ( (f N U ) 1 (w) ), R(w) ) = max w K f N (L) χ(ϕ(w), R(w)) < ε max χ(ϕ(w), R(w)) w f N (L) and thus C f N,U (R Ω ) V ε,l,u (h). Altogether, it follows that C f N,U (R Ω ) is contained in C f N,U (V ε,k,ω (g)) V ε,l,u (h). This shows the topological transitivity of the sequence (C f n,u ). Remark 2.3. The proof of the above theorem runs similarly as the proof of Theorem 2.2 in [11]. However, in contrast to the holomorphic setting, we now do not have to take care of the position of the poles of the rational function R. Moreover, the proof shows that the sequence (C f n,u ) is actually topologically mixing (for a definition, see e.g. [10, Definition 1.56]). In the sequel we restrict ourselves to the case Ω = C, that is, we ask for the existence of universal functions which are meromorphic in C. The first application of Theorem 2.2 deals with the case of a symbol f which is holomorphic near an attracting fixed point, which we suppose to be 0, i.e. we have f(0) = 0 and 0 < λ < 1 for λ := f (0). According to G. Kœnigs linearization theorem, there exist open neighborhoods U and V of 0 as well as 4

5 a conformal mapping ϕ : U V which conjugates the map f U : U U to the linear function F : V V, F (w) := λw, i.e. ϕ f n = F n ϕ = λ n ϕ holds on U for all n N (see e.g. [5, Theorem II.2.1]). In particular, we obtain ϕ(0) = 0 and locally uniform convergence f n 0 on U. Since F, ϕ and ϕ 1 are injective, the same is also true for f U = ϕ 1 F ϕ. Hence, the restriction f U : U U is injective and Theorem 2.2 implies Corollary 2.4. Comeager many functions in M (C ) are M (U )-universal for C f. Remark 2.5. In the special case V = U = C and f = F we obtain that the operator C f is topologically transitive on M (C ) (and even topologically mixing; cf. Remark 2.3). It is easily seen that whenever we consider an invariant open set U with 0 U and so that f is not injective on U, there cannot exist a function g M (U) which is M (U)-universal for C f, that is C f : M (U) M (U) cannot be universal. In particular, this is the case for punctured open neighborhoods of superattracting fixed points of the symbol however, at least in the holomorphic setting, then C f fulfills a certain universality property on many small compact subsets of such a neighborhood (cf. [11], Theorem 3.5). Let now f be a holomorphic self-map on a domain D and let A := A f,0 := {z D : f n (z) 0} denote the basin of attraction of 0 under f. It is easily seen that A is completely invariant under f and open, this is, f H(A) and f(a) A as well as f 1 (A) A (see e.g. [5, p. 28]). It is well-known that the function ϕ from above can be extended to a holomorphic function Φ on the whole basin of attraction A and that the equation Φ f = λ Φ still holds on A (see e.g. [5, p. 32]). Moreover, it can be shown that this property determines the function Φ up to a multiplicative nonzero constant (cf. [5, p. 28]). We consider the backward orbit O := O f,0 := n N {z D : f n (z) = 0} of 0 under f and we write A := A \ O. Again, A is completely invariant under f and open but f is in general not injective on A. Our aim is to determine the set of limit functions of the sequence (g f n A ) for generic functions g M (C ). For U A open we write ω(u, g, f) for the set of all spherically locally uniform limit functions of the sequence (g f n U ). Theorem 2.6. Let f H(D) be a self-map and so that n N f n (D) is a neighborhood of 0. If g M (A ) is M (U )-universal for some open neighborhood U n N f n (D) of 0 on which f is conjugated to w λw then ω(a, g, f) = {h Φ A : h M (Φ(A ))}. 5

6 Proof: Combining the statements of Corollary 4.4, Lemma 4.5 and Lemma 4.6 in [11], the proof runs exactly in the same way as the proof of the corresponding result in the holomorphic setting (cf. [11, Theorem 4.7]). One only has to observe the following three points: Due to the assumption that U is contained in the 0-neighborhood n N f n (D), one can prove analogously to the proof of Lemma 4.3 in [11] (which is need for the proof of Corollary 4.4) that we have ω ( f n (U), g, f ) { h Φ f n (U) : h M (Φ(f n (U))) }, n N 0. Moreover, in view of Corollary 2.4, it now suffices to consider the punctured neighborhood U instead of the set U 0 which was constructed in the proof of Lemma 4.5. ii) in [11]. Finally, an analogous statement as that of Lemma 4.5 in [11] now also holds in the meromorphic setting. Corollary 2.4 shows that a neighborhood U of 0 exists so that comeager many g M (C ) turn out to be M (U )-universal for C f. From Theorem 2.6 we obtain Corollary 2.7. Under the assumption of Theorem 2.6, comeager many g M (C ) enjoy the property that ω(a, g, f) = {h Φ A : h M (Φ(A ))}. Remark 2.8. The assumption of n N f n (D) being a neighborhood of 0 is obviously satisfied in the case that f(d) = D. Moreover, according to Picard s theorem it is also satisfied for entire functions f (with D = C). Remark 2.9. For entire functions f, Theorem 2.6 yields an analogous statement in the spherical setting as Theorem 4.7 in [11] does in the holomorphic setting. The main difference lies in the fact that we can make a statement about the generic set of limit functions on the whole basin of attraction of 0 under f except the minimal (countable) set O. In contrast, the exceptional sets in the holomorphic setting are larger (unions of rectifiable curves with Hausdorff dimension 1) and cannot chosen to be minimal. 3 A Geometric Approach in Case of Rational Symbols of Degree Two Theorem 2.6 implies that M (U)-universal functions g M (C ) exist for all U A with the property that Φ U is injective. In this section, it is our aim to determine parts of A having this property in case that the symbol is a rational function of a special form. We consider a rational function f of degree d f 2 which has an attracting fixed point at the origin. As described before Corollary 2.4, there exist open neighborhoods U and V of 0 as well as a conformal map ϕ : U V which conjugates the map f U : U U to the linear function w f (0) w on V. 6

7 Moreover, there exists a maximal radius r max > 0 such that the inverse function ϕ 1 can be extended to a conformal function on the open disk V max := {z C : z < r max } (see e.g. [12, Lemma 8.5]). Hence, there exists a maximal open neighborhood U max of 0 and a conformal map ϕ max : U max V max which conjugates f Umax : U max U max to the linear function w f (0) w on V max (cf. the proof of Lemma 8.5 in [12]). Corollary 2.4 yields that comeager many functions in M (C ) are M (Umax)-universal for C f. In order to prove M (W )-universality of C f for suitable open sets W which go beyond U max, we need more precise information about f on the whole basin of attraction. Again, the concept of conformal conjugation will be the main tool here. 3.1 Finite Blaschke Products The Fatou set F = F f of a rational function f is defined as the set of all points z C for which there exists a neighborhood U of z such that {f n U : n N} is a normal family in M (U). Considering a simply connected invariant component G of F f with G C, G C, the Riemann mapping theorem implies the existence of a conformal map ψ : G D := {z C : z < 1}. Hence, ψ conjugates f G : G G to the function g := ψ f ψ 1 : D D (cf. [7, p. 586]). Since rational functions map components of their Fatou sets properly onto each other (see e.g. [15, Theorem 1 on p. 39]), we obtain that g is a proper self-map of D (i.e. g 1 (K) is compact for each compact set K D). It is known that each proper self-map of D is the restriction to D of a finite Blaschke product, i.e. of a rational function B : C C of the form B(z) := e iθ d k=1 z α k 1 α k z for θ R, d N and α 1,... α d D (see e.g. [15, Exercise 6 on p. 7], or [14, p. 185]). If, in the above situation, there exists exactly one k {1,..., d} with α k = 0, it is known that the Fatou set of B equals D C \D and that the origin and the point at infinity are attracting fixed points of B (cf. [12, Problem 7-b on p. 70]). Hence, the classification theorem of Fatou components (see e.g. [3, p. 163]) implies that we have locally uniform convergence B n 0 on D and B n on C \D. In particular, B D is a surjective holomorphic self-map on D and the basin of attraction A of 0 under B D equals D. In order to determine open subsets U of A such that there exist M (U)- universal functions for C B in M (C ), we now consider the simplest subclass of finite Blaschke products having an attracting fixed point at the origin. Due to the above considerations, this subclass is given by the functions B α : C C, B α (z) := z ϕ α (z) = z 7 z α 1 αz, α D.

8 From now on, we fix some α D and we write B := B α. Calculating B (z) = 0, we see that there exist exactly two critical points of B which are given by z 1 := 1 1 α 2 α and z 2 := α 2. α Another simple calculation yields 0 < z 1 < α < 1 < 1/ α < z 2 so that z 1 is the only critical point of B in D. Because of arg α = arg(1/α), the points 0, z 1, α, 1/α and z 2 lie on the same straight line through 0. Moreover, the two distances z 1 1/α and z 2 1/α are equal. For z, w C, a short computation shows that we have B(z) = B(w) if and only if w = z or w = ϕ α (z). For z C, we compute that we have ϕ α (z) = z if and only if z = z 1 or z = z 2. Thus, the critical points of B are exactly the fixed points of ϕ α. Indeed, the map ϕ α is a key factor in understanding the dynamics of B because its geometry on D is known. In order to describe this geometry, we introduce the following notations: For a straight line L in the complex plane, we denote by R L the reflection map which reflects each point in C with respect to L. Considering a point w C, a radius r > 0 and the closed disk C := {z C : z w r}, we define I C (z) for z C\{w} as the point which lies on the ray {w + t(z w) : t 0} and which has distance r 2 / z w to w. The map I C is called the inversion on C. We have I C (C \{w}) = C\C and I C (C\C) = C \{w}. Each point on C is a fixed point of I C. If w = 0 and r = 1, we have I D (z) = 1/z for all z C. For a closed disk D which is orthogonal to C (i.e. the two tangents to D through the two points in C D pass through w), it follows I C (D) = D (see [13, p. 149]). Considering the finite Blaschke product B = B α, now let L be the straight line which passes through 0 and α. As stated above, we have z 1, z 2, 1/α L and 0 < z 1 < α < 1 < 1/ α < z 2 as well as z 1 1/α = z 2 1/α (see Figure 1). Moreover, the two critical points z 1 and z 2 of B are exactly the two fixed points of ϕ α. Therefore, there exists a closed disk C with center 1/α which is orthogonal to D with I C (α) = 0 and C L = {z 1, z 2 } and such that ϕ α acts on D as the composition of the reflection R L and the inversion I C in any order, i.e. we have ϕ α D = (R L I C ) D = (I C R L ) D (1) (see [13, p. 207]). We define the disjoint subsets S 1, S 2, S 3 and S 4 of D as illustrated in Figure 2. According to (1), we see that ϕ α interchanges S 1 and S 3 as well as S 2 and S 4, i.e. we have ϕ α (S 1 ) = S 3, ϕ α (S 3 ) = S 1, ϕ α (S 2 ) = S 4 and ϕ α (S 4 ) = S 2. Lemma 3.1. The finite Blaschke product B is injective on the unions S k S l for all k, l {1, 2, 3, 4} with {k, l} / {{1, 3}, {2, 4}}. Proof: Let z, w S 1 S 2 with B(z) = B(w). Due to the above considerations, it follows that z = w or ϕ α (z) = w. But the latter cannot be true because in this case we would obtain w = ϕ α (z) ϕ α (S 1 S 2 ) = ϕ α (S 1 ) ϕ α (S 2 ) = S 3 S 4, 8

9 Figure 1 Figure 2 a contradiction. Hence, B is injective on S 1 S 2. According to the geometry of ϕ α, the injectivity of B on the other unions can be shown similarly. In order to extend the universality statement of Corollary 2.4 in case of the finite Blaschke product B, we now have a look at the contour lines of B. For 0 < r < 1 and z D, we have B(z) = r if and only if z z α = r 1 αz. (2) Ignoring the factor 1 αz on the right-hand side, the points z C fulfilling the equation z z α = r would form a Cassini oval. In general, for two points w 1, w 2 C and a constant c > 0, the Cassini oval C(w 1, w 2, c) is defined as the set of all points in the complex plane having the property that the product of their distances to w 1 and w 2 has constant value c 2, i.e. C(w 1, w 2, c) := { z C : z w 1 z w 2 = c 2}. 9

10 The shapes of these sets according to the value of c are well-known. For small positive values of c, the set C(w 1, w 2, c) consists of two disjoint Jordan curves which look like small circles around w 1 and w 2. Increasing c, these two components become more and more egg-shaped until they meet each other in the middle of the line segment between w 1 and w 2 for c = w 1 w 2 /2. The figureeight-shaped set L(w 1, w 2 ) := C(w 1, w 2, w 1 w 2 /2) is a lemniscate. For larger values c > w 1 w 2 /2, the Cassini ovals C(w 1, w 2, c) consist of one component which first looks like a sand glass, then like an ellipse and finally like a large circle (cf. [13, p. 71f.]). In equation (2), the factor 1 αz on the right-hand side acts as an error term so that the sets C(0, α, r) := {z D : z z α = r 1 αz } look like deformed Cassini ovals. As z 1 is a critical point of B, the deformed lemniscate L(0, α) is reached for r = B(z 1 ), i.e. we have L(0, α) := {z D : B(z) = B(z 1 ) }. Let W 1 and W 2 be the components of the open set {z D : B(z) < B(z 1 ) } which contain 0 and α, respectively. On the left-hand side of Figure 3, a plot of several deformed Cassini ovals is displayed in case of α = i. The right-hand side of Figure 3 shows a schematic plot of the deformed lemniscate L(0, α) and the sets W 1 and W 2. Figure 3 Lemma 3.2. W 1 is invariant under B and we have B(W 2 ) W 1. Proof: For z W 1, it follows that B(B(z)) B(z) < B(z 1 ). Hence, we obtain B(z) W 1 W 2 and thus B(W 1 ) W 1 W 2. We have 0 = B(0) 10

11 B(W 1 ), and the continuity of B and the connectedness of W 1 imply that B(W 1 ) is also connected. According to B(z) B(W 1 ), there exists a path in B(W 1 ) W 1 W 2 which connects 0 and B(z). As this is not possible for B(z) W 2, we obtain B(z) W 1. Because of 0 = B(α) B(W 2 ), the inclusion B(W 2 ) W 1 can be proved in exactly the same way. Theorem 3.3. Comeager many functions in M (C ) are M (W 1 )-universal and M (W 2 \{α})-universal for C B. Proof: We denote by G 1 the set of all functions in M (C ) which are M (W 1 )- universal for C B and by G 2 the set of all functions in M (C ) which are M (W 2 \{α})-universal for C B. i) Lemma 3.1 and Lemma 3.2 imply that B W 1 : W1 W1 is injective. As we have locally uniform convergence B n W 1 0 (cf. the end of the first paragraph of this subsection), Theorem 2.2 yields that G 1 is comeager in M (C ). ii) Firstly, we show by induction that B n is injective on W 2 for all n N. For n = 1, this follows from Lemma 3.1. Now, let B n be injective on W 2 and let z, w W 2 with B n+1 (z) = B n+1 (w), i.e. B(B n (z)) = B(B n (w)). As we have B n (z), B n (w) W 1 by Lemma 3.2 and as B is injective on W 1 due to Lemma 3.1, we obtain B n (z) = B n (w) and hence z = w. Considering the open sets D := W1 (W 2 \{α}) and U := W 2 \{α} D, Lemma 3.2 yields B(D) W1 D. According to the injectivity of all iterates B n on U and the locally uniform convergence B n U 0, Theorem 2.2 yields that G 2 is comeager in M (C ). Remark 3.4. Corollary 2.4 and Theorem 2.6 imply the existence of open subsets U of A such that there exist M (U)-universal functions for C B in M (C ). In general, we do not have any information on the shape or the exact location of these sets. However, the geometric approach of proving the statement of Theorem 3.3 yields the two concrete sets W1 and W 2 \{α} which support universal functions for C B in M (C ). 3.2 Rational Functions of Degree Two According to the considerations at the beginning of the Subsection 3.1, it is now our aim to transfer the universality statements of the previous subsection to the situation of arbitrary rational functions having an attracting fixed point. In order to do so, we consider a rational function f of degree d f 2 and a simply connected component G of F f with G C, G C, which contains the origin as an attracting fixed point of f. Then G is invariant under f so that there exists a conformal map ψ : G D which conjugates f G : G G to the restriction B D : D D of a finite Blaschke product B of degree d B N of the form d B 1 B(z) = e iθ z α k z 1 α k z, z C, k=1 11

12 where θ R and α 1,..., α db 1 D (cf. Subsection 3.1 and [2, Exercise on p. 109]). Hence, we have ψ f = B ψ and thus (ψ f) f = (B ψ) ψ. (3) In order to apply the results of Subsection 3.1, we now want to find situations in which we have d B = 2. The following lemma provides a necessary and sufficient condition: Lemma 3.5. In the above situation, let G further be completely invariant under f. Then we have d B = 2 if and only if d f = 2. Proof: First, let d f = 2. Because of f (0) 0, there exists a point z C with f( z) = f(0) G. The backward invariance of G under f implies z f 1 (G) G, and due to d f = 2, we have f ( z) 0. As ψ is injective, we obtain ψ(0) ψ( z), and according to the above conjugation, it follows that B(ψ(0)) = ψ(f(0)) = ψ(f( z)) = B(ψ( z)). Because of f (0) 0 and f ( z) 0, (3) yields B (ψ(0)) 0 and B (ψ( z)) 0. Assuming that there exists a point w D\{ψ(0), ψ( z)} with B(w) = B(ψ(0)), the conjugation of f G to B D would imply that f ( ψ 1 (w) ) = ψ 1( B(w) ) = ψ 1( B(ψ(0)) ) = ψ 1( ψ(f(0)) ) = f(0). Due to d f = 2 and f(0) = f( z), we would obtain ψ 1 (w) {0, z} and hence w {ψ(0), ψ( z)}, a contradiction. Thus, we have B 1({ B(ψ(0)) }) = {ψ(0), ψ( z)}. According to B (ψ(0)) 0 and B (ψ( z)) 0, this yields d B = 2. On the other hand, now let d f 3. As above, the backward invariance of G under B yields the existence of a point w 0 G\{0} with f(w 0 ) = f(0). In case of f (w 0 ) 0, there must be another point w 1 G\{0, w 0 } with f(w 1 ) = f(0). Due to the injectivity of ψ, the three points ψ(0), ψ(w 0 ) and ψ(w 1 ) are distinct, and the above conjugation implies that they are all mapped onto B(ψ(0)) under B so that we have d B 3. If f (w 0 ) = 0, (3) yields B (ψ(w 0 )) = 0. As we have ψ(0) ψ(w 0 ) and B(ψ(0)) = B(ψ(w 0 )), it follows again that d B 3. Corollary 3.6. Let f be a rational function of degree two and let G C, G C, be a simply connected component of F f which is completely invariant under f and which contains an attracting fixed point of f. Then there exists some α D such that f G is conjugated to B α D. Proof: According to Lemma 3.5, there exist θ R and β D as well as a conformal map ψ : G D which conjugates f G to B θ,β, where B θ,β : D D, B θ,β (z) := e iθ z z β 1 βz. 12

13 Putting α := e iθ β D and considering the rotation ψ θ : D D, ψ θ (z) := e iθ z, the following equation holds for all z D: ψ θ (B θ,β (z)) = e iθ B θ,β (z) = e iθ e iθ z = e iθ z z β 1 βz = eiθ z e iθ z α 1 α e iθ z = B α(e iθ z) = B α (ψ θ (z)). e iθ z e iθ β 1 (e iθ α) z Thus, ψ θ conjugates B θ,β to B α D so that ψ θ ψ conjugates f G to B α D. Example 3.7. Let P be a polynomial of degree two and let the origin be an attracting fixed point of P. Then the component G(0) of F P containing 0 and the component G( ) of F P containing the superattracting fixed point of P both are completely invariant under P (see e.g. [5, p. 103]). In this situation, G(0) and G( ) must be simply connected ([5, Theorem IV.1.1]). Thus, G(0) fulfills the assumptions of Corollary 3.6. Examples of polynomials P of this kind are given by the family of quadratic polynomials P λ : C C, P λ (z) := z 2 + λz, where λ D (cf. [15, p. 40]). Moreover, there are examples of rational, non-polynomial functions in the situation of Corollary 3.6. Indeed, for each 0 < b < 1, the rational function z R b : C C, R b (z) := z 2 bz + 1, has an attracting fixed point at b and a neutral fixed point at 0. Furthermore, we have / F Rb, and the component G(b) of F Rb containing b as well as the component of F Rb containing an attracting petal at 0 are completely invariant under R b (see [5, p. 83]) and hence simply connected ([5, Theorem IV.1.1]). Thus, G(b) fulfills the assumptions of Corollary 3.6. Now, we transfer the universality statements of Subsection 3.1 to the situation of a rational function f of degree two and a simply connected, completely invariant component G C, G C, of F f which contains the origin as an attracting fixed point of f. According to Corollary 3.6, there exists an α D, a finite Blaschke product B α of the form B α (z) := z(z α)/(1 αz), z C, and a conformal map ψ : G D such that the equation ψ f = B α ψ holds on G. Now, let z 1 be the critical point of B α in D (cf. the considerations at the beginning of Subsection 3.1) and let V 1 := ψ 1 (W 1 ) and V 2 := ψ 1 (W 2 ), where W 1 and W 2 are the components of the open set {z D : B α (z) < B α (z 1 ) } which contain 0 and α, respectively. In view of the above conjugation, Theorem 3.3 implies Theorem 3.8. Comeager many functions in M (C ) are M (V 1 )-universal and M (V 2 \ {ψ 1 (α)})-universal for C f. 13

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