IINTERPOLATION OF POWER MAPPINGS
JACK BURKART AND KIRILL LAZEBNIK
Abstract.
Let ( M j ) ∞ j =1 ∈ N and ( r j ) ∞ j =1 ∈ R + be increasing sequences satisfying somemild rate of growth conditions. We prove that there is an entire function f : C → C whosebehavior in the large annuli { z ∈ C : r j · exp( π/M j ) ≤ | z | ≤ r j +1 } is given by a perturbedrescaling of z (cid:55)→ z M j , such that the only singular values of f are rescalings of ± r M j j . Wedescribe several applications to the dynamics of entire functions. Contents
1. Introduction 12. Preliminaries 43. Interpolation of Power Maps 64. Entire Functions 145. A Dynamical Application 17References 201.
Introduction
The problem of constructing analytic functions with prescribed geometry arises in areasacross function theory. One general approach consists of defining a convergent infinite prod-uct, which has the advantage, for instance, of yielding an explicit expression for the zerosof the function. We mention several dynamical applications of this approach: the first con-struction of an entire function with a wandering domain [Bak76], constructions of entirefunctions of slow growth with Julia set = C [BE00], in the study of escaping sets of en-tire functions [RS13], the construction of transcendental entire functions with -dimensionalJulia sets [Bis18], and in the dynamics of sine maps [ERS20].A general principle used in the above works is that the behavior of an infinite productis dominated within certain regions by certain factors in the product. Our main resultsimilarly proves the existence of an entire function whose behavior is dominated withincertain annuli by certain prescribable monomials. One advantage of our approach, however,is a precise description of the singular values of the entire function. This information is ingeneral difficult to glean from an infinite product, and is often crucial in applications. Beforedescribing further motivation and applications, we state our main result. First we will need a r X i v : . [ m a t h . C V ] J a n JACK BURKART AND KIRILL LAZEBNIK the following definition, which will serve as an assumption on the growth rate of the degreeof the aforementioned monomials.
Definition 1.1.
Let ( M j ) ∞ j =1 ∈ N be increasing, and ( r j ) ∞ j =1 ∈ R + . We say that ( M j ) ∞ j =1 , ( r j ) ∞ j =1 are permissible if r j +1 ≥ exp (cid:0) π (cid:14) M j (cid:1) · r j for all j ∈ N , r j j →∞ −−−→ ∞ , and sup j M j +1 M j < ∞ . (1.1)With Definition 1.1, we can state our main result: Theorem A.
Let ( M j ) ∞ j =1 , ( r j ) ∞ j =1 be permissible, r := 0 and c ∈ C (cid:63) := C \ { } . Set (1.2) c := c , and c j := c j − · r M j − − M j j − for j ≥ . Then there exists an entire function f : C → C and a homeomorphism φ : C → C such that f ◦ φ ( z ) = c j z M j for r j − · exp( π/M j − ) ≤ | z | ≤ r j , j ∈ N . (1.3) Moreover, if (cid:80) ∞ j =1 M − j < ∞ , then | φ ( z ) /z − | → as z → ∞ . The only singular values of f are the critical values ( ± c j r M j j ) ∞ j =1 .Remark . The homeomorphism φ is a K -quasiconformal homeomorphism (see Definition2.1), where K depends only on sup j ( M j +1 /M j ) . The conclusion | φ ( z ) /z − | → is deducedfrom the Teichmüller-Wittich-Belinskii Theorem (see Theorem 2.6), and in many applica-tions, we can even deduce uniform estimates || φ ( z ) /z − || L ∞ ( C ) < ε (see Section 5). Wefurther remark that a precise description of the critical points and zeros of f are also given(see Proposition 4.9 and Remark 4.10), up to the perturbation φ . The “scaling” constants ( c j ) ∞ j =1 ensure that z (cid:55)→ c j z M j and z (cid:55)→ c j +1 z M j +1 both map | z | = r j to the same scale.Lastly, we comment that if we replace the condition r j → ∞ of Definition 1.1 with thecondition r j → r ∞ < ∞ , we obtain a result similar to Theorem A, but with the domain of f equal to r ∞ · D rather than C (see Theorem B in Section 4).As indicated in Remark 1.2, our methods rely on quasiconformal surgery , a collection oftechniques to which we refer to [FH09] for a survey. Among these techniques, there are atleast two distinct approaches both termed quasiconformal surgery . The first consists in theconstruction of a quasiregular function g possessing a g -invariant Beltrami coefficient µ . Theintegrating map for µ then conjugates g to a holomorphic function. This approach appearsfirst in [Sul85] (to the best of the authors’ knowledge), and is the most common use of theMeasurable Riemann Mapping Theorem in complex dynamics, as it is inherently dynamical.A different approach also termed quasiconformal surgery consists of constructing a quasireg-ular function g which in turn yields a holomorphic function g ◦ φ − , where φ is the integratingmap for the Beltrami coefficient g z /g z . This is the approach used in the present work, and haslong found fundamental application in the type problem and value distribution theory (see,for instance, [Dra86], [Ere86]), and, surprisingly, has found recent application in complex NTERPOLATION OF POWER MAPPINGS 3 (A) (B) (C) (D)
Figure 1.
Illustrated is an interpolation, defined on ≤ | z | ≤ , betweenthe identity on T and the angle-doubling map on T . In (A), the white (re-spectively, black) vertices on the outer circle represent preimages under − (respectively, +2) of the angle doubling map, and on the inner circle we rep-resent − (respectively, +1 ) by a white (respectively, black) vertex.dynamics despite the lack of a conjugacy between g and g ◦ φ − . Indeed, this approach wasused in [Bis15] in settling a long-standing question about the existence of wandering domainsfor functions with bounded singular set (see also [FGJ15], [FJL19], [Laz19], [MPS20]).A general difficulty in this approach lies in proving that φ is only a small perturbationof the identity, which may be deduced, for instance, from showing that g is holomorphic(and hence g z /g z = 0 ) except on a very small set. To this end, the work [Bis15] introduceda technique termed quasiconformal folding , which has found many recent applications incomplex dynamics and in function theory more broadly (see, for instance, [Bis14], [BP15],[Rem16], [OS16], [BRS17], [Laz17], [BL19], [AB20]). Our main methods of proof for TheoremA are influenced by this technique, but we emphasize that the present work does not relydirectly on the results of [Bis15], and indeed may be read and understood independently ofthe aforementioned works.To conclude the Introduction, we remark on several applications of Theorem A and presenta sketch of the proof. In the present manuscript, we briefly present in Section 5 how TheoremA yields a robust approach to the construction of entire functions with multiply connectedwandering domains (first constructed in [Bak76] - see also [Bak85], [Hin94], [KS08], [BZ11],[Ber11], [BRS13]). In a companion manuscript [BL], we will show how Theorem A givesa different approach to the result of [Bis18] on existence of transcendental entire functionswith -dimensional Julia set (see also [Bur19]).The proof of Theorem A consists of defining a quasiregular function g by z (cid:55)→ c j z M j inthe large annuli r j − · exp( π/M j − ) ≤ | z | ≤ r j , and finding an efficient interpolation betweenthese power mappings in the thin annuli r j ≤ | z | ≤ r j · exp( π/M j ) . We sketch the main ideain the simple case of interpolation in ≤ | z | ≤ between z (cid:55)→ z on T := { z : | z | = 1 } and z (cid:55)→ z on T . We will refer to Figure 1. It will suffice to interpolate between the identity JACK BURKART AND KIRILL LAZEBNIK (A) (B) (C) (D)
Figure 2.
Illustrated is the main interpolation used in Theorem A in aslightly more elaborate case than in Figure 1: interpolation between z (cid:55)→ z and z (cid:55)→ z . The map (B) (cid:55)→ (C) is given by θ → θ .on T and the angle doubling map on T . The map (A) (cid:55)→ (B) is the identity on T andpiecewise-linear in logarithmic coordinates, except on the red segment where it is -valuedas illustrated (the inverse (B) (cid:55)→ (A) is called the folding map in [Bis15]). The map (B) (cid:55)→ (C)is given by angle-doubling. The composition (A) (cid:55)→ (C) is still -valued on the red segment,whence the map (C) (cid:55)→ (D) identifies these values on the real-axis. Although useful for asketch of the proof, we remark that we do not actually use a separate map (C) (cid:55)→ (D) toidentify these values, but rather this procedure is built into the definition of (B) (cid:55)→ (C) (seeDefinition 3.11).The more general case of interpolation between ( z (cid:55)→ z n ) | T and ( z (cid:55)→ z m ) | r T , r > ,for general n < m is similar (see Figure 2): we attach radial segments (illustrated in red)to each preimage of − under z (cid:55)→ z n on T which play a similar role as above, and use ingeneral a more elaborate placement of vertices along these radial segments. A careful analysisshows that the desired interpolation can be achieved in an annulus of modulus / n , withdistortion constants depending only on the ratio m/n and not otherwise on n , m . Thisessentially yields the first conclusion in Theorem A, and the statement about critical valuesfollows by an inspection of which points the above quasiregular function branches at.We now briefly outline the paper. After collecting a couple preliminary results we willneed in Section 2, we detail the specifics of the main interpolation in Section 3 where theprimary contributions of the present work are contained. Section 4 applies the results ofSection 3 to build the entire function of Theorem A. In Section 5, we consider dynamicalapplications of Theorem A. 2. Preliminaries
Definition 2.1.
An orientation-preserving homeomorphism φ : C → C is said to be a quasiconformal mapping if φ ∈ W , loc ( C ) and || φ z /φ z || L ∞ ( C ) ≤ k for some ≤ k < . NTERPOLATION OF POWER MAPPINGS 5
Remark . We refer the reader to [LV73] for a detailed study of quasiconformal mappings.We will assume a familiarity with the basic theory in what follows. We remark that a quasiregular mapping is one which may be represented by a composition f ◦ φ , where f isholomorphic, and φ is quasiconformal (see [LV73], Chapter VI). Notation 2.3.
We abbreviate piecewise linear by PWL. Given a quasiregular map f , wewill denote the dilatation constant of f by K ( f ) , where ≤ K ( f ) < ∞ . We denote k ( f ) := ( K ( f ) − / ( K ( f ) + 1) , and occasionally we will use the notation A ( r , r ) := { z ∈ C : r < | z | < r } .We record a Theorem due to Teichmüller, Wittich, and Belinskii (Theorem 2.6 below).As already mentioned in the Introduction, we will use this result to deduce the conclusion | φ ( z ) /z − | → of Theorem A. The statement of the result is taken from Theorem 6.1 of[LV73], to which we refer for the relevant bibliography. We note that in Theorem 2.6, d A ( z ) refers to area measure. Before stating the result, we recall two definitions which will appearin the Theorem. Definition 2.4.
Let ψ : C → C be a quasiconformal mapping. The dilatation quotient of ψ is defined by(2.1) D ( z ) := | ψ z ( z ) | + | ψ z ( z ) || ψ z ( z ) | − | ψ z ( z ) | . The quantity D ( z ) is defined for a.e. z for a quasiconformal mapping ψ (see [LV73], SectionIV.1.5). Definition 2.5.
Let ψ be a quasiconformal mapping defined in a neighborhood of a point z . The map ψ is said to be conformal at z if the limit lim z → z ψ ( z ) − ψ ( z ) z − z (2.2)exists, in which case we denote the limit by ψ z ( z ) . Theorem 2.6.
Let ψ be a K -quasiconformal mapping of the finite plane onto itself with ψ (0) = 0 and (2.3) I ( r ) := 12 π (cid:90) | z | This Section contains the primary technical contributions of the present work. We willdescribe in detail the interpolation procedure sketched in the Introduction and prove therelevant estimates. We begin by describing the first step of the interpolation in logarithmiccoordinates. Definition 3.1. We define a region W := { z ∈ C : 0 < Re ( z ) < } \ { z ∈ C : 0 ≤ Re ( z ) ≤ / and Im ( z ) ∈ Z + 1 } . Given m ∈ N , we also define a triangulation T m of W as follows (see Figure 3). Place verticesat(3.1) , (cid:18) jm · i (cid:19) mj =0 , and (cid:18) i + j m − (cid:19) m − j =0 . Label the vertices black or white as follows: and are black, i is white, and the othervertices are colored so that adjacent vertices on Re ( z ) = 1 or ( Im ( z ) = 1) ∩ ( Re ( z ) ≤ / have different colors. There is a triangulation of W ∩ [0 , formed by connecting each vertexwith real part ≤ / to i · ( m − /m . Iteratively reflecting this triangulation of W ∩ [0 , along a subset of horizontal lines Im ( z ) = k , k ∈ Z defines the triangulation T m . Remark . The coloring of the vertices in Definition 3.1 is not essential, however it is a usefulconvention: as already discussed in the Introduction, the colored vertices represent preimagesof ± under a map defined below in Definition 3.11, with the two colors corresponding tothe two choices ± . Definition 3.3. Given m ∈ N , we define a triangulation T (cid:48) m of { z ∈ C : 0 < Im ( z ) < } asfollows (see Figure 3). First place vertices at(3.2) (cid:18) jm · i (cid:19) mj =0 and (cid:18) jm · i (cid:19) mj =0 . We color and black, and require adjacent vertices with the same real part to have differentcolors. There is a triangulation of [0 , defined by connecting each vertex on Re ( z ) = 0 to i · ( m − /m . Iteratively reflecting this triangulation of [0 , along a subset of thehorizontal lines Im ( z ) = k , k ∈ Z defines the triangulation T (cid:48) m . Definition 3.4. We define a PWL-homeomorphism(3.3) ψ m : W → { z ∈ C : 0 < Im ( z ) < } NTERPOLATION OF POWER MAPPINGS 7 ψ ψ Figure 3. Illustrated is the definition of ψ m in W ∩ [0 , in the cases m = 2 , .as follows. We specify that ψ m (0) := 0 and ψ m (1 + jm · i ) := 1 + jm · i for ≤ j ≤ m , and(3.4) ψ m (cid:18) i + j m − (cid:19) := j + 1 m · i for ≤ j ≤ m − . (3.5)As the triangulations T m ∩ [0 , and T (cid:48) m ∩ [0 , are compatible, (3.4) and (3.5) uniquelydetermine a triangulation-preserving PWL-homeomorphism(3.6) ψ m : W ∩ [0 , → [0 , . We extend the map ψ m to a PWL-homeomorphism ψ m : W → { z ∈ C : 0 < Im ( z ) < } byrepeated applications of the Schwarz-reflection principle. Proposition 3.5. For any m ∈ N , the map (3.7) ψ m : W → { z ∈ C : 0 < Im ( z ) < } JACK BURKART AND KIRILL LAZEBNIK of Definition 3.4 is quasiconformal.Proof. The map(3.8) ψ m : W ∩ [0 , → [0 , is defined as a PWL map, and hence the dilatation constant of (3.8) is the supremum of thedilatations of the m +2 R -linear maps in the definition of (3.8). Thus (3.8) is quasiconformal.As the definition of ψ m in W is obtained by the Schwarz-reflection principle, it follows that ψ m is also quasiconformal with the same constant. (cid:3) Remark . K ( ψ m ) → ∞ as m → ∞ .The above essentially defines the first step in our interpolation in logarithmic coordinates.We now revert back to the z -plane (see also Figure 5). Definition 3.7. We define a planar region E n := exp (cid:16) πn · W (cid:17) . Given n , m ∈ N , we also define a map(3.9) η n,m ( z ) := (cid:16) z (cid:55)→ exp (cid:16) πn · z (cid:17)(cid:17) ◦ ψ m ◦ (cid:16) z (cid:55)→ nπ log z (cid:17) for z ∈ E n . Proposition 3.8. For any m ∈ N , the map (3.10) η n,m : E n → { z ∈ C : 1 ≤ | z | ≤ exp( π/n ) } is a quasiconformal homeomorphism. Moreover, K ( η n,m ) depends only on m .Proof. This follows from (3.9) and Proposition 3.5. (cid:3) Having transferred the first step of our interpolation back to the z -plane, we now havea function η n,m which is the desired map (A) (cid:55)→ (B) of the Introduction. The composition(A) (cid:55)→ (C) is given by ( z (cid:55)→ z nm ) ◦ η n,m . As discussed in the Introduction, ( z (cid:55)→ z nm ) ◦ η n,m is -valued across the radial arcs on the boundary of E n , and we will need a procedure to“identify” these 2 values. This is given in Definition 3.11 below, for which we will first needthe following: Definition 3.9. Following [BL19], we define a -quasiconformal map(3.11) σ : { z ∈ C : | z | > } → C \ [ − , as follows (see Figure 4). Denote by µ ( z ) := ( z + 1) / ( z − the Möbius transformationmapping | z | > conformally to the right-half plane. Let ν denote the -quasiconformal mapsending the right-half plane to C \ ( −∞ , that is the identity on | arg( z ) | ≤ π/ and triplesangles in the remaining sector. Then(3.12) σ := µ ◦ ν ◦ µ. NTERPOLATION OF POWER MAPPINGS 9 Figure 4. Illustrated is the map σ of Definition 3.9. This Figure is takenfrom [BL19]. Remark . We will denote X := supp ( σ z /σ z ) (illustrated as the dark gray region in theleft-most copy of C in Figure 4). Note that σ is the identity on { z ∈ C : | z | > } \ X . Definition 3.11. Given n , m ∈ N , we define a quasiregular map g n,m on the region E n asfollows. Consider first the map(3.13) ( z (cid:55)→ z mn ) ◦ η n,m : E n → { z ∈ C : 1 ≤ | z | ≤ exp( mπ ) } . Note that X ⊂ { z ∈ C : 1 ≤ | z | ≤ exp( mπ ) } for any m ∈ N . There is a partition of the boundary of E n into the two circular arcs | z | = 1 , | z | = exp( π/n ) and n radial arcs perpendicular to | z | = 1 (see Figure 5 for the case n = 2 where the radialarcs are colored red). The preimage of X under (3.13) consists of mn components: let U denote the union of those components which neighbor one of the n radial arcs on theboundary of E n . We define(3.14) g n,m ( z ) := (cid:40) σ ◦ ( z (cid:55)→ z mn ) ◦ η n,m ( z ) z ∈ U ( z (cid:55)→ z mn ) ◦ η n,m ( z ) z ∈ E n \ U. Remark . Note that the two formulas in (3.14) agree on ∂U as σ ( z ) = z for z ∈ ∂X . Proposition 3.13. For any n , m ∈ N , The map g n,m of Definition 3.11 is quasiregular on ≤ | z | ≤ exp( π/n ) . Moreover, K ( g n,m ) depends only on m , and: (1) g n,m ( z ) = z n if | z | = 1 , and (2) g n,m ( z ) = z mn if | z | = exp( π/n ) .Proof. It is evident from Definition 3.11 that g n,m is quasiregular in E n as it is a compositionof quasiregular maps in E n . Thus to show that g n,m is quasiregular on ≤ | z | ≤ exp( π/n ) ,it suffices (by a standard removability result) to show that g n,m extends continuously acrossthe radial arcs on the boundary of E n . For z on such a radial arc, ( z (cid:55)→ z mn ) ◦ η n,m ( z ) is η , z (cid:55)→ z (A) (B) (C) Figure 5. The two dark-gray regions in (C) represent the set X of Remark3.10, and the dark-gray regions in (B) and (A) represent the pullback of X under the maps z (cid:55)→ z and η , ◦ ( z (cid:55)→ z ) , respectively. The definition of g , differs from that of ( z (cid:55)→ z ) ◦ η , only in the four dark-gray regions in (A)neighboring the radial arcs colored red. The map ( z (cid:55)→ z ) ◦ η , is -valuedon these arcs, whereas g , is single-valued and indeed continuous across thesearcs. -valued with both values lying at complex-conjugate points on T , whence σ identifies thesetwo points. Thus g n,m extends continuously across the radial arcs on the boundary of E n .It remains to show (1) and (2). (2) is evident since | z | = exp( π/n ) is disjoint from U , and η n,m ( z ) = z for | z | = exp( π/n ) (see Definition 3.4 of ψ m ). The verification of(1) is straightforward: it follows from the definition of η n,m and fact that for | z | = 1 , g n,m ( z ) = ( z (cid:55)→ z mn ) ◦ η n,m ( z ) . We leave the details to the reader. That K ( g n,m ) dependsonly on m follows from the Formula (3.14) and Proposition 3.8. (cid:3) In order to understand the singular values of the function f of Theorem A, we will needto keep track of those points at which our interpolating function is locally n : 1 for n > .To this end, we introduce the following definition: Definition 3.14. Let g be a quasiregular function, defined in a neighborhood of a point z ∈ C . We say that z is a branched point of g if for any neighborhood U of z , the map g | U is n : 1 onto its image for n > . If, further, n = 2 , we say that z is a simple branched point.We say w ∈ C is a branched value of g if w = g ( z ) for a branched point z of g . Proposition 3.15. Let g be as in Definition 3.11. Define g ( z ) := z n for | z | < , and g ( z ) := z mn for | z | > exp( π/n ) . Then the only non-zero branched points of g are: (3.15) exp (cid:18) πn · l (2 m − 2) + i · π (2 k − n (cid:19) for ≤ l ≤ m − , and ≤ k ≤ n. NTERPOLATION OF POWER MAPPINGS 11 Moreover, all non-zero branched points of g are simple, and the only non-zero branched valuesof g are ± .Remark . The expression (3.15) is simply a formula for the black and white verticeslocated on the radial arcs colored red in, for instance, Figure 5(A), but excluding thosevertices at the outer-most tips of the radial arcs. Proof. We first note that the extended formula defines a quasiregular function on C byremovability of analytic arcs for quasiregular mappings. It is readily verified then from thedefinition of g that any neighborhood of each of the points (3.15) is mapped onto itsimage by g . The points (3.15) are sent to ± by g , where we note that the sign may bedetermined by the coloring of the vertex as described in Remark 3.2. Lastly, again from thedefinition of g , one verifies directly that there are no remaining non-zero branched points of g . (cid:3) We will also need to record the zeros of our interpolating function for later application.The formulas are listed below in Proposition 3.17, but they are readily seen to just be themidpoint between each adjacent black/white vertex on a radial segment pictured in, forinstance, Figure 5(A). Proposition 3.17. Let g be as in Proposition 3.15. Then the zeros of g are given by: and exp (cid:18) πn · l + m − i · π (2 k − n (cid:19) for ≤ l ≤ m − and ≤ k ≤ n, (3.16) all of which are simple except for which is of multiplicity n .Proof. Recalling Definition 3.9 of σ , we see that σ − (0) = {± i } . Let S := ( z (cid:55)→ z mn ) − ( ± i ) .Then g − (0) is readily seen to be those points x such that η n,m ( x ) = {± i } and x ∈ U for U as in Definition 3.11. Such x are listed in (3.16). (cid:3) We now generalize our interpolating function slightly to allow for interpolation between z (cid:55)→ z n and z (cid:55)→ z M for M > n , where M is not necessarily a multiple of n . While there isa necessary level of complication in the formulas in the following proof, the idea is simple.Denoting m := (cid:98) M/n (cid:99) , we will use η n,m in some parts of the interpolating annulus, and η n,m +1 in others, so that we add the necessary amount M − n of new vertices (branchedpoints). Next, we adjust by a homeomorphism τ of the circle which sends the n old verticesand M − n new vertices on T together to span the M th roots of unity, and finally post-composewith an adjustment of the map z (cid:55)→ z M as in Definition 3.11. Proposition 3.18. Let n , M ∈ N with M > n . Then there exists a quasiregular function (3.17) g : { z ∈ C : 1 ≤ | z | ≤ exp( π/n ) } → { z ∈ C : | z | ≤ exp( M π/n ) } , such that: (1) g ( z ) = z n if | z | = 1 , (2) g ( z ) = z M if | z | = exp( π/n ) , and (3) K ( g ) depends only on M/n , and not otherwise on M or n .Proof. Consider the region E n as in Definition 3.7. Let m := (cid:98) M/n (cid:99) . Let(3.18) E n,j := E n ∩ (cid:26) z ∈ C \ { } : 2( j − πn ≤ arg( z ) ≤ jπn (cid:27) for ≤ j ≤ n. Let p := n − M + nm , and recall the mapping η n,m as in Definition 3.7, where we use theconvention η n, ( z ) ≡ z . We define the mapping(3.19) η ( z ) := (cid:40) η n,m ( z ) z ∈ ∪ pj =1 E n,j η n,m +1 ( z ) z ∈ ∪ nj = p +1 E n,j . It is readily verified that η : E n → { z ∈ C : 1 ≤ | z | ≤ exp( π/n ) } is a homeomorphism, since ψ m , ψ m +1 are both the identity on the lines y = 2 Z . Next, we define a homeomorphism τ : T → T by(3.20) τ ( θ ) := (cid:40) nmM · θ θ ∈ [0 , pπ/n ] ( m +1) nM · θ θ ∈ [ − π ( n − p ) /n, . Extend τ to a self-homeomorphism of A (1 , exp( π/n )) by linearly interpolating τ | T with theidentity on | z | = exp( π/n ) . Since n − p = ( M/n ) / ( M/n − m ) , it is readily seen from (3.20)that K ( τ ) depends only on M/n , and not otherwise on M or n .Finally, we define g by appropriately adjusting Definition 3.11. Namely, let X := supp ( σ z /σ z ) as in Remark 3.10. Then the pullback of X under(3.21) ( z (cid:55)→ z M ) ◦ τ ◦ η : E n → A (1 , exp( M π/n )) consists of M components. Denoting by U the union of those M − n ) components whichneighbor a radial arc on the boundary of E n , we again define(3.22) g ( z ) := (cid:40) σ ◦ ( z (cid:55)→ z M ) ◦ η ( z ) z ∈ U ( z (cid:55)→ z M ) ◦ η ( z ) z ∈ E n \ U. The proof that g satisfies (1)-(3) then is the same as in Proposition 3.13. (cid:3) Following Proposition 3.15, we will also list the branched points, branched values, andzeros of the map g of Proposition 3.18. But first, we describe a simple rescaling which allowsour annular region of interpolation to have an inner boundary lying on a circle of variableradius. NTERPOLATION OF POWER MAPPINGS 13 Proposition 3.19. Let r > , c ∈ C (cid:63) and n , M ∈ N with M > n . There exists a quasiregularfunction (3.23) g : { z ∈ C : r ≤ | z | ≤ r exp( π/n ) } → { z ∈ C : | z | ≤ cr n exp( M π/n ) } , such that: (1) g ( z ) = c · z n if | z | = r , (2) g ( z ) = c · z M /r M − n if | z | = r exp( π/n ) , and (3) K ( g ) depends only on M/n , and not otherwise on M , n , r or c .Proof. We first consider the case when M = m · n for m ∈ N . Let(3.24) E n,r := exp (cid:16) πn · W + log r (cid:17) . By Proposition 3.8, the map(3.25) η n,m,r := η n,m ◦ ( z (cid:55)→ z/r ) : E n,r → { z ∈ C : 1 ≤ | z | ≤ exp( π/n ) } is a quasiconformal homeomorphism where K ( η n,m,r ) depends only on m . Following Defini-tion 3.11, we can define a map ˜ g in E n,r by adjusting the values of ( z (cid:55)→ z mn ) ◦ η n,m,r in aneighborhood of the radial arcs on the boundary of E n,r . The proof that ˜ g is quasiregularon { z ∈ C : r ≤ | z | ≤ r exp( π/n ) } is then the same as in Proposition 3.13. Set g ( z ) := c · r n · ˜ g ( z ) . (3.26)Then the proof that g satisfies (1)-(3) similarly follows as in Proposition 3.13. Lastly, thecase when M > n is not necessarily a multiple of n follows from the same adjustment of theabove interpolation as in Proposition 3.18. (cid:3) Remark . When we wish to emphasize the dependence of g on the parameters n , M , r , c , we will write g n,M,r,c (in that order).Lastly, we record the branched points, branched values, and zeros of our interpolating func-tion. The same proofs as for Propositions 3.15 and 3.17 prove the following: Proposition 3.21. Let g , and notation be as in Proposition 3.19. Set m := (cid:98) M/n (cid:99) , and p := n − M + nm . Define g ( z ) := cz n for | z | < r , and g ( z ) := cz M /r M − n for | z | > r exp( π/n ) .Then the only non-zero branched points of g are: r · exp (cid:18) πn · l (2 m − 2) + i · π (2 k − n (cid:19) for ≤ l ≤ m − and ≤ k ≤ p , and (3.27) r · exp (cid:18) πn · l m + i · π (2 k − n (cid:19) for ≤ l ≤ m − , and p + 1 ≤ k ≤ n. Moreover, all non-zero branched points of g are simple, and the only non-zero branched valuesof g are ± cr n . The zeros of g are given by the multiplicity n zero at , and the simple zerosdescribed by replacing l with l + 1 / in the expressions (3.27). Entire Functions With the technical work of Section 3 behind us, we will now apply our interpolationrepeatedly between power maps of increasing degree in “increasing” annuli in the plane. Aslong as the degree of the power maps increases by at most a fixed constant factor in eachconsecutive annulus, this procedure gives a function quasiregular in C . We formalize thisbelow. Definition 4.1. Let c ∈ C (cid:63) , ( M j ) ∞ j =1 ∈ N be increasing, and ( r j ) ∞ j =1 ∈ R + . We set r := 0 , M := 1 . Suppose that(4.1) r j +1 ≥ exp (cid:0) π (cid:14) M j (cid:1) · r j for all j ∈ N , and r j j →∞ −−−→ ∞ . Set(4.2) c := c , and c j := c j − · r M j − − M j j − = c · j (cid:89) k =2 r M k − − M k k − for j ≥ . We then define:(4.3) h ( z ) := (cid:40) c j · z M j if r j − · exp( π/M j − ) ≤ | z | ≤ r j g M j ,M j +1 ,r j ,c j ( z ) if r j ≤ | z | ≤ r j · exp( π/M j ) . over all j ∈ N . Remark . By Proposition 3.19, the two definitions in (4.3) agree on | z | = r j . ByProposition 3.19 and the definition (4.2) of c j , the two definitions in (4.3) agree on | z | = r j · exp( π/M j ) . Thus the formula (4.3) determines a well-defined function h : C → C . Definition 4.3. We will say that ( M j ) ∞ j =1 ∈ N , ( r j ) ∞ j =1 ∈ R + are weakly permissible if (4.1)is satisfied. Remark . Suppose ( M j ) ∞ j =1 , ( r j ) ∞ j =1 are weakly permissible. Then, according to Definition1.1, we have that ( M j ) ∞ j =1 , ( r j ) ∞ j =1 are permissible if and only if the sequence ( M j /M j − ) ∞ j =1 is bounded. Remark . The map h of Definition 4.1 is determined by a choice of c ∈ C (cid:63) and weaklypermissible ( M j ) ∞ j =1 , ( r j ) ∞ j =1 . Proposition 4.6. Suppose ( M j ) ∞ j =1 , ( r j ) ∞ j =1 are weakly permissible, and c ∈ C (cid:63) . Then thefunction h as defined in Definition 4.1 is quasiregular on compact subsets of C . If, moreover, ( M j ) ∞ j =1 , ( r j ) ∞ j =1 are permissible, then h is quasiregular on C and K ( h ) depends only on sup j ( M j /M j − ) and not otherwise on ( M j ) ∞ j =1 , ( r j ) ∞ j =1 .Proof. In each annular region in the Definition (4.3), the map h is either analytic, or quasireg-ular by Proposition 3.19. By removability of analytic arcs, the map h is therefore quasireg-ular across the boundaries of the annular regions. As any compact subset of C meets only NTERPOLATION OF POWER MAPPINGS 15 finitely many such annular regions, it follows that h is locally quasiregular. If the sequence ( M j /M j − ) ∞ j =1 is bounded (or equivalently ( M j ) ∞ j =1 , ( r j ) ∞ j =1 are permissible), then each ofthe maps g M j ,M j +1 ,r j ,c j ( z ) have a dilatation bounded uniformly over j (with a bound depend-ing only on sup j ( M j /M j − ) ) by (3) of Proposition 3.19, and so the last statement of theProposition follows. (cid:3) The definition of permissible thus ensures that the formula (4.3) defines a quasiregularfunction, in which case we may integrate the Beltrami coefficient h z /h z to obtain a quasi-conformal map φ such that h ◦ φ − is holomorphic. If, moreover, the M j increase sufficientlyquickly (see (4.4)), the total region of interpolation is sufficiently small to guarantee confor-mality of φ at ∞ . This is the content of Theorem 4.8 below, which will be presented afterthe following definition: Definition 4.7. Let ( M j ) ∞ j =1 , ( r j ) ∞ j =1 be permissible. We say that ( M j ) ∞ j =1 , ( r j ) ∞ j =1 are stronglypermissible if also(4.4) (cid:88) j M − j < ∞ . Theorem 4.8. Let ( M j ) ∞ j =1 , ( r j ) ∞ j =1 be strongly permissible, and c ∈ C (cid:63) . Then there exists aquasiconformal mapping φ : C → C such that f := h ◦ φ − as in Theorem A is holomorphic,and (4.5) (cid:12)(cid:12)(cid:12)(cid:12) φ ( z ) z − (cid:12)(cid:12)(cid:12)(cid:12) z →∞ −−−→ . Proof. The proof is an application of Theorem 2.6. The existence of a quasiconformal φ : C → C such that h ◦ φ − is holomorphic and φ (0) = 0 follows from applying the MeasurableRiemann Mapping Theorem to h z /h z . Let(4.6) ψ ( z ) := 1 /φ (1 /z ) . Then ψ is a quasiconformal self-mapping of C satisfying ψ (0) = 0 . Let K denote thequasiconformal constant of ψ , and take r < ∞ . We calculate I ( r ) := 12 π (cid:90) | z | Let ( M j ) ∞ j =1 , ( r j ) ∞ j =1 be permissible, c ∈ C (cid:63) , and f , φ as in Theorem 4.8.Set m j := (cid:98) M j /M j − (cid:99) , and p j := M j − − M j + M j − m j . Then the only critical points of f are and the simple critical points given by φ (cid:18) r j · exp (cid:18) πM j − · l j m j − i (2 k j − πM j − (cid:19)(cid:19) , and (4.13) φ (cid:18) r j · exp (cid:18) πM j − · l (cid:48) j m j + i (2 k (cid:48) j − πM j − (cid:19)(cid:19) , where j ∈ N , and ≤ k j ≤ p j , ≤ l j ≤ m j − , and p j + 1 ≤ k (cid:48) j ≤ M j − , ≤ l (cid:48) j ≤ m j − .The only singular values of f are the critical values ( ± c j r M j j ) ∞ j =0 .Proof. By Proposition 3.21, the only branched points of h are and the points given by(4.13) without the φ factor. Thus, as f := h ◦ φ − , it follows the only critical points of f are φ (0) = 0 and those given in (4.13). Again, as f := h ◦ φ − , by Proposition 3.21 each of thepoints in (4.13) is mapped to ± c j r M j j as j ranges over N . There are no asymptotic values of f : if γ → ∞ is a curve, then f ( γ ) is unbounded. (cid:3) NTERPOLATION OF POWER MAPPINGS 17 Remark . The same proof as for Proposition 4.9 shows that the zeros of f are given by (multiplicity M ) and the simple zeros whose formulae is given by replacing l j and l (cid:48) j inthe expressions in (4.13) by l j + 1 / , l (cid:48) j + 1 / , respectively.Theorem A now follows: Proof of Theorem A: We take f := h ◦ φ − as in Theorem 4.8. The conclusions of TheoremA are then included in the statements of Theorem 4.8 and Proposition 4.9.The applications of the present manuscript of which the authors are aware are in regardsto entire functions with an essential singularity at ∞ , and so it is natural to impose theassumption r j → ∞ in Definition 1.1 of “permissibility”. However, this is inessential to mostof our arguments. Indeed, if we assume instead that r j → r ∞ < ∞ , we can define h exactlyas in (4.3), and then the argument of Proposition 4.6 together with an application of theMeasurable Riemann Mapping theorem yields the following: Theorem B. Let c ∈ C (cid:63) , < r ∞ < ∞ , ( M j ) ∞ j =1 ∈ N be increasing, ( r j ) ∞ j =1 ∈ R + such that r j +1 ≥ exp (cid:0) π (cid:14) M j (cid:1) · r j for all j ∈ N , r j j →∞ −−−→ r ∞ , and sup j M j +1 M j < ∞ . (4.14) Set r := 0 and (4.15) c := c , and c j := c j − · r M j − − M j j − for j ≥ . Then there exists a holomorphic function f : r ∞ D → C and a K -quasiconformal homeomor-phism φ : r ∞ D → r ∞ D with K depending only on sup j ( M j +1 /M j ) such that: f ◦ φ ( z ) = c j z M j for r j − · exp( π/M j − ) ≤ | z | ≤ r j , j ∈ N . (4.16) Moreover, the only singular values of f are the critical values ( ± c j r M j j ) ∞ j =1 . Missing from the statement of Theorem B is an application of criteria for conformalityat a point (in Theorem A the criteria is (cid:80) j M − j < ∞ , and the point is ∞ ). Although wedo not record them here, one can produce analogous criteria in Theorem B for conformalityof φ at points on | z | = r ∞ . Indeed, we may extend φ to a self-map of ˆ C by the Schwarzreflection principle, and apply Theorem 2.6 at a point lying on | z | = r ∞ .5. A Dynamical Application In this Section, we discuss some brief dynamical applications of Theorem A. We will discussan approach to constructing entire functions with multiply-connected wandering domainsusing Theorem A. But first we will show how, in certain applications, we can conclude muchmore than conformality at ∞ of the map φ in Theorem A. Namely we can conclude a uniformestimate || φ ( z ) /z − || L ∞ ( C ) < ε in the following situation: given a function f = h ◦ φ − as inTheorem A, if we define h n by replacing h in | z | ≤ r n with z (cid:55)→ c n z M n , we obtain a sequence of entire functions f n = h n ◦ φ − n with f n ≈ f in | z | ≥ r n . As n → ∞ , the maps φ n convergeuniformly (in the spherical metric on ˆ C ) to the identity so that || φ n ( z ) /z − || L ∞ ( C ) < ε for large n . This argument is very useful in dynamical applications: it will be used in thecompanion paper [BL], and a related argument is used in [Bis15], [FJL19], [Laz19]. Weformalize the above discussion below. Definition 5.1. Let ( M j ) ∞ j =1 , ( r j ) ∞ j =1 be strongly permissible, and c ∈ C (cid:63) . (5.1)Denote by f = h ◦ φ − the entire function obtained by applying Theorem A to (5.1), where h is quasiregular and φ quasiconformal. We define a sequence ( f n ) ∞ n =0 of entire functions asfollows. Let h := h and for n ≥ : h n ( z ) = (cid:40) c n z M n if | z | ≤ r n h ( z ) if | z | ≥ r n . Define f n := h n ◦ φ − n , where φ n : C → C is the unique quasiconformal mapping such that:(1) f n is holomorphic,(2) φ n (0) = 0 , and(3) | φ n ( z ) /z − | → as z → ∞ . Remark . That we may normalize φ n in Definition 5.1 so as to satisfy (3) requires justi-fication. To this end, let(5.2) I n ( r ) := 12 π (cid:90) | z | Let ( M j ) ∞ j =1 , ( r j ) ∞ j =1 be strongly permissible, c ∈ C (cid:63) , and notation as inDefinition 5.1. Then for any ε > , there exists N ε ∈ N such that for n ≥ N ε , || φ n ( z ) /z − || L ∞ ( C ) < ε. (5.4) Proof. Briefly, this is a consequence of the error term in the conclusion (2.4) of Theorem2.6 depending only on K and I , and not otherwise on the quasiconformal mapping underconsideration. Let us explain further. We follow the notation of Remark 5.2. For I n as in(5.2), we have already noted that I n ( r ) ≤ I ( r ) . Moreover, K ( ψ n ) ≤ K ( ψ ) by (5.3). Thus,by Theorem 2.6, there is a function ι : R + → R + with(5.5) (cid:12)(cid:12)(cid:12)(cid:12) ψ n ( z ) z − (cid:12)(cid:12)(cid:12)(cid:12) < ι ( | z | ) NTERPOLATION OF POWER MAPPINGS 19 where ι ( r ) → as r → , and ι does not depend on n . Thus, given ε > , there exists R > such that:(5.6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ n ( z ) z − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ( | z |≥ R ) < ε for all n ∈ N . On the other hand, for all z ∈ C we have D n ( z ) → as n → ∞ . Thus, by a standardargument, we have(5.7) || φ n ( z ) − z || L ∞ ( | z |≤ R ) n →∞ −−−→ . The result now follows from (5.6), (5.7), and a Köbe distortion estimate for φ (cid:48) n (0) . (cid:3) We now turn to an application: we will show that as long as our parameters satisfycertain simple relations, the resulting entire function of Theorem A has a multiply connectedwandering domain. Notation 5.4. For < α < , we let A αj := { z ∈ C : α − r j − · exp( π/M j − ) < | z | < αr j } . Theorem 5.5. Suppose ( M j ) ∞ j =1 , ( r j ) ∞ j =1 are strongly permissible, c ∈ C (cid:63) , and (5.8) r j +1 = c j r M j j . Then, for any α > , there exists N α such that, for j ≥ N α , A αj is contained in a wanderingdomain for f as in Theorem A.Remark . The condition (5.8) is sufficient to guarantee a wandering domain for the func-tion f of Theorem A, but is far from necessary. Proof. As usual, we denote f = h ◦ φ − . By Theorem A, there exists N = N α ∈ N such that(5.9) | φ ( z ) /z − | < |√ α − | for | z | ≥ r N . It follows that(5.10) φ (cid:0) A αj (cid:1) ⊂ A √ αj for j ≥ N. By (5.8), we have h ( r j ) = r j +1 for all j . Together with expansivity of h , this implies:(5.11) f (cid:0) A αj (cid:1) ⊂ A αj +1 for j ≥ N, perhaps after increasing N . Thus the iterates of f converge uniformly to ∞ on A αj for j ≥ N ,and so each such A αj is contained in a Fatou component for f , which we will call Ω j .If we suppose by way of contradiction that Ω j = Ω k for some j (cid:54) = k , then Ω j = Ω j +1 , andthis would imply that Ω j is unbounded. But Ω j is multiply connected since f (0) = 0 , andthis is a contradiction since all multiply connected Fatou components of f must be boundedby [Bak75, Theorem 1]. Thus, we may conclude that the (Ω j ) ∞ j = N form a distinct sequenceof Fatou components. Since f (Ω j ) ⊂ Ω j +1 , it follows that each such Ω j is a wanderingcomponent for f . (cid:3) Remark . Let notation be as in Theorem 5.5, and consider the family f n of Definition 5.1.It is not difficult to show using Theorem 5.3 that if < α < , then for sufficiently large n we have that each A αj for j ∈ N is contained in a wandering component Ω j of f n . References [AB20] Simon Albrecht and Christopher J. Bishop. 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