Virtually repelling fixed points

Publ. Mat. 47 (2003), 195–209 VIRTUALLY REPELLING FIXED POINTS Xavier Buff Abstract In this article, we study the notion of virtually repelling fixed point. We first give a definition and an interpretation of it. We then prove that most proper holomorphic mappings f:U→Vwith U contained in Vhave at least one virtually repelling fixed point. 1. Preliminaries Let f:(C,α)→(C,α)beaholomorphic germ fixing α. Definition 1. The multiplicity mof αas a fixed point of fis the residue m= residue 1−f(z) z−f(z)dz, α. In other words, it is the multiplicity of αasarootofz−f(z). When m=1,αis a simple fixed point and when m>1, αis a multiple fixed point. The multiplier of fat αis the derivative λ=f(α). When |λ|>1, αis repelling, when |λ|<1, αis attracting and when |λ|=1,αis indifferent. If αis repelling or multiple, it is weakly repelling. Remark. Observe that αis multiple if and only if λ=1. One easily proves that analytic conjugacy preserves the multiplier at a fixed point and that topological conjugacy preserves the property of being repelling, attracting or indifferent. Topological conjugacy preserves the multiplier at an indifferent fixed point (see [N]or[PM]). It also preserves the multiplicity of a fixed point and two germs having multiple fixed points with the same multiplicity are always topologically conjugate (see [C]). 2000 Mathematics Subject Classification. 37F99. Key words. Fixed point, holomorphic index, rational-like mapping. 196 X. Buff Definition 2. The residue fixed point index ι(f,α)offat a fixed point αis the residue ι(f,α)=residue 1 z−f(z)dz, α. If the multiplicity is mand (ι(f,α)) <m/2, αis virtually repelling,if (ι(f,α)) >m/2, αis virtually attracting and if (ι(f,α)) = m/2, αis virtually indifferent. It is known that the residue fixed point index is invariant under analytic conjugacy (see [M, Lemma 12.3]) but not under topological conjugacy (as mentioned above, two germs having a multiple fixed point with the same multiplicity are always topologically conjugate). Since the notions introduced above are invariant under analytic conjugacies, it makes sense to talk about the multiplier or the residue fixed point index of a holomorphic germ f:(U, α)→(U, α) where Uis an arbitrary Riemann surface and α∈Uis an arbitrary point (for example the point ∞in P1). In Section 2, we show that a virtually repelling fixed point with multiplicity mmay be thought as the superposition of mfixed points which are repelling on average. More precisely, we prove the following theorem. Theorem 1. A germ f:(C,α)→(C,α)has a virtually repelling fixed point at αif and only if any sufficiently small perturbation fεof fhas at least one virtually repelling fixed point close to α. Besides, if αhas multiplicity mand is not virtually repelling, there exist arbitrarily small perturbations fεhaving mattracting fixed points close to α. In [J], Jellouli proves that when P(z)=e2iπp/qz+z2, then q+1 2−2q−1≤(ι(P◦q,0)) ≤q+1 2. The upper bound says that 0 is a virtually repelling fixed point of P◦q. The proof relies on the fact that the multiplicity of 0 as a fixed point of P◦qis q+1and that there exist small perturbations Pεof Psuch that P◦q εhas only repelling fixed points. In [Sh3], [B] and [BE], refinements of this result are given. In Sections 3 and 4, we prove that most ramified coverings f:U→V with U⊂Vhave at least one virtually repelling fixed point. Definition 3. Let U⊂Vbe Riemann surfaces and f:U→Vbe a holomorphic map. We say that fis repelling on average if and only if f has finitely many fixed points αk∈Uwith multiplicities mkand ι(f,αk)<1 2mk. Virtually Repelling Fixed Points 197 When a holomorphic map f:U→Vis repelling on average, the barycenter of the quantities ι(f,αk)/mkweighted with multiplicities mk has real part less than 1/2. Therefore, one of those quantities must have real part less than 1/2. Hence, fmust have at least one virtually repelling fixed point. Let us first consider the case where U=V=P1, i.e., the case of rational maps. It is well-known (see [M, Section 10]) that for any rational map f:P1→P1,wehave the equality  {α∈P1|α=f(α)} ι(f,α)=1. This is known as the Fatou’s Index Formula, or the Holomorphic Fixed Point Formula. When the degree of fis d≥2, it has d+1fixed points counted with multiplicity. Therefore fis repelling on average and has at least one virtually repelling fixed point. In Sections 3 and 4, we generalize this result. In Section 3, we introduce the notion of rational-like mappings (see [R1] and [R2]). Those are ramified coverings f:U→Vwhere U and Vare planar Riemann surfaces with finite Euler characteristic and U is relatively compact in V.Weprove an analog of Douady-Hubbard’s Straightening Theorem (see [DH]). We then prove the following result. Theorem 2. If f:U→Vis a rational-like mapping, then it has at least one virtually repelling fixed point. Besides, if Vis simply connected, f is repelling on average. Question. Is a rational-like mapping always repelling on average? In Section 4, we still consider the case where f:U→Vis a ramified covering with U⊂V.Weallow Unot to be compactly contained in V, but we restrict to the case where Vis simply connected. In that case, f:U→Vis conformally conjugate to a ramified covering g:U→D (via an isomorphism ϕ:V→D). Therefore, we may restrict our study to ramified coverings f:U→Dwith Ucontained in D. Theorem 3. Let f:U→Dbeaproper holomorphic map of degree d≥2 with Ucontained in D.If|f(z)−z|is bounded away from zero as z∈U tends to ∂U, then, fhas dfixed point in U,counting multiplicities, and fis repelling on average. In particular, it has at least one virtually repelling fixed point. Finally, in Section 5, we give some applications. In particular, we show that if fis a rational map with a (super)attracting fixed point α whose immediate basin Ωαis not simply connected, then Ωαseparates 198 X. Buff two virtually repelling fixed points of f.Wealso show that when a rational map fhas a fixed Herman ring A, each connected component of P1\Acontains a virtually repelling fixed point. 2. Perturbations of virtually repelling fixed points In this section we try and understand the notion of virtually repelling fixed points. First, observe that this notion is finer than the one of weakly repelling fixed point. Indeed, when αis not a multiple fixed point, then the residue fixed point index ι(f,α) and the multiplier λare related by ι(f,α)= 1 1−λ. Consequently, |λ|>1ifand only if (ι(f,α)) <1/2 and a simple fixed point is virtually repelling if and only if it is repelling. When αis a multiple fixed point of multiplicity mand (ι(f,α)) < m/2, we think of αhas being the superposition of mfixed point which are repelling on average. This is essentially the content of Theorem 1 which says that a germ f:(C,α)→(C,α) has a virtually repelling fixed point at αif and only if any sufficiently small perturbation fεof fhas at least one virtually repelling fixed point close to α.Inparticular, if all the fixed point of fεare simple, at least one of them is repelling. Proof of Theorem 1: First, the result is clear if αis a simple fixed point. Now, assume αis a multiple fixed point and let mbe its multiplicity. The multiplicity and the residue fixed point index of fat αcan be defined via the integrals of (1 −f(z))/(z−f(z)) and 1/(z−f(z)) along a small loop γturning once around α. Those integrals depend continuously on f. It follows that for any sufficiently small perturbation fεof f, the sum of the multiplicities mkat the fixed point αkof fεcontained in the region delimited by γis equal to mand the sum of residue fixed point indices of fεat the points αkis close to the residue fixed point index of fat α. In particular, for any sufficiently small perturbation fε,wehave ι(fε,α k)<1 2mk. This precisely means that the barycenter of the quantities ι(fε,α k)/mk weighted with multiplicities mkhas real part less than 1/2. In particular one of those quantities has real part less than 1/2 and the corresponding fixed point is virtually repelling. Conversely, assume αis not virtually repelling. We will show that there exist arbitrarily small perturbations fεhaving mattracting fixed points close to α.Wewill obtain the perturbation in two steps. Virtually Repelling Fixed Points 199 Step 1: If αis virtually indifferent, we first make a perturbation that turns it into a virtually attracting fixed point. For example, consider the perturbation fε:(C,α)→(C,α) defined by 1 z−fε(z)=1 z−f(z)+ε z−α. The map fεhas a fixed point of multiplicity mat αand the residue fixed point index is ι(fε,α)=ι(f,α)+ε. When (ε)>0, the fixed point becomes virtually attracting. Step 2: Without loss of generality, we may now assume that αis virtually attracting, i.e., (ι(f,α)) >m/2. In order to get a hand on the residue fixed point index, we will use the following fact, which is known but not absolutely obvious (see for example the appendix in [BE]): one may perform an analytic change of coordinates so that the Taylor expansion of fat αbecomes f(z)=α+(z−α)+(z−α)m+ι(f,α)(z−α)2m−1+O(|z−α|2m). Let us first study the case of the polynomial g:C→Cdefined by g(z)=α+(z−α)+(z−α)m+ι(f,α)(z−α)2m−1. We define λεby 1 1−λε =i ε+ι(f,α) m and set gε=α+λε(g−α). The fixed points of the polynomials gεare the solutions of the equation 1+(z−α)m−1+ι(f,α)(z−α)2(m−1) =1 λε . For small values of ε, the polynomial gεhas mattracting fixed points close to α: one at αwith multiplier λεand α1,...,α m−1which, by symmetry, all have the same multiplier. Thus, the residue fixed point index ι(gε,α k)doesnotdepend on k=1,...,m−1. Now, we have ι(gε,α)= i ε+ι(f,α) m and ι(gε,α)+ m−1  k=1 ι(gε,α k)−→ ε→0ι(f,α). 200 X. Buff Therefore, for all k=1,...,m−1, we have ι(gε,α k)= −i (m−1)ε+ι(f,0) m+o(1). In particular, for small values of ε, the residue fixed point indices have real part close to (ι(f,α))/m > 1/2 and the fixed points are attracting. We will now define the perturbation of f. Let us work in a local coordinate where f(z)=g(z)+O(|z−α|2m−1). We define the perturbation fε by 1 z−fε(z)=1 z−gε(z)+1 z−f(z)−1 z−g(z). The fixed point of fεare the poles of 1/(z−fε(z)). Since 1 z−f(z)=1 z−g(z)+O(1) the fixed points of fεwhich are close to αand their residue fixed point indices coincide with the fixed point of gεwhich are close to αand their residue fixed point indices. In particular, for small values of ε,fεhas m attracting fixed points close to α. Remark. The proof given above also shows that when f:(C,α)→(C,α) has a virtually repelling fixed point at αwith multiplicity m, there exist arbitrarily small perturbations fεhaving mrepelling fixed points close to α. 3. Rational-like mappings Definition 4. A rational-like mapping is a proper holomorphic mapping f:U→Vof degree d≥2, where Uand Vare connected open subsets of P1with finite Euler characteristic and Uis relatively compact in V. A rational-like mapping comes with a filled-in Julia set Kf(the set of non-escaping points), and a Julia set Jf(the boundary of Kf). The Julia set Jfmay equivalently be defined as the closure of the set of repelling periodic points. The following result shows that rational-like mappings behave like rational maps. This result is probably not new and may already appear somewhere in the literature. Theorem 4 (Straightening Theorem).For any rational-like map f:U→ V, there exist a rational map F:P1→P1, neighborhoods Uand Vof the Julia set JFand a quasi-conformal homeomorphism ϕ:V→V which conjugates f:U→Vto F:U→V. Virtually Repelling Fixed Points 201 Remark. The ∂derivative of ϕmay be chosen to vanish everywhere on Kf. Proof: We will only sketch the main lines of the proof of Theorem 4. The proof mimics the one by Douady and Hubbard of the straightening theorem for polynomial-like mappings (see [DH]). By restricting Vif necessary, we may assume that Uand Vhave smooth boundaries. We denote by (Bi)i∈I the connected components of P1\Vand by (Dj)j∈J the connected components of P1\U. The sets Iand Jare finite by assumption, and the sets Biand Djare Jordan domains. Besides, each Biis contained in a unique Dj, and the inclusion induces a map ι:I→J. The rational-like map finduces a mapping f∗:J→Iso that f(∂Dj)=∂Bf∗(j). We may now define an extension of fto the Riemann sphere. This extension will be quasi-regular and will satisfy Shishikura’s principle (see [Sh1, Lemma 1]). Therefore, the proof will be completed. If (j0,j 1,...,j n)isaperiodic cycle of the map ι◦f∗,wedefine Ajk to be the annulus Djk\Bf∗(jk−1)and we let djkbe the degree of f| ∂Djk.Wechoose a real number r∈]0,1[, and we define A jkto be the annulus {z∈C|rdk−1≤|z|≤r}.Wechoose quasi-conformal homeomorphisms ϕjk:Ajk→A jkwhich satisfy ϕjk◦f=(ϕjk−1)djk−1 on ∂Djk−1. Then, we use ϕjkto glue the dynamics of z→ zdjkin each disk Djk. This defines the extension of fin every disk Djsuch that j is a periodic point of the map ι◦f∗.For the remaining disks Dj,we choose any quasi-regular extension. Since any rational map Fhas at least one weakly repelling fixed point, and since this point is contained in JF,itfollows that any rational-like map has at least one weakly repelling fixed point. However, since quasiconformal conjugacies do not necessarily preserve the property of being virtually repelling, one has to work a little to prove Theorem 2. Proof of Theorem 2: Let us first prove that any rational-like mapping has at least one virtually repelling fixed point. If this were not the case, we could find a rational-like mapping f:U→Vhaving only virtually attracting and virtually indifferent fixed points. The idea is to find a perturbation fε:U→Vwhich is still rational-like but the fixed points of which are all attracting, which clearly gives a contradiction. Denote by α1,...,α n∈Uthe fixed points of f.Inaneighborhood of each fixed point αk, Theorem 1 provides a local perturbation fε,k which only has attracting fixed points. The only difficulty consists in gluing the perturbations fε,k into a global one. For any meromorphic 202 X. Buff function h:U→P1,wecan write a decomposition h=P(h)+R(h), where P(h)isthe polar part of h(i.e., sum of negative powers of (z−pj) at the poles pj), and R(h)∈O(1) is the regular part of h. Then, in a neighborhood of each fixed point αk, the polar part P(1/(z−fε,k)) converges to the polar part P(1/(z−f)) at αkas εtends to 0. Therefore, we can define a global perturbation fε:U→P1by 1 z−fε = n  k=1 P1 z−fε,k +R1 z−f(z). The perturbation fεis defined on the whole set Ubut a priori, it is not rational-like. However, since fεconverge to fon every compact subset of Uas εtends to 0, we can find a restriction Vof Vso that U⊂V⊂Vand fε:U=f−1 ε(V)→Vis rational-like. The fixed points of fε:U→Vand their residue fixed point indices coincide with the fixed points in Uof the maps fε,k and their residue fixed point indices. By construction, all those points are attracting which gives the required contradiction. Let us now prove that any rational-like mapping f:U→V, with V simply connected, is repelling on average. In that case, f:U→V is conjugate, via an isomorphism ϕ:V→D,toarational-like mapping g:U→D. The number of fixed points of fand gare the same, and the residue fixed point indices coincide. Thus, without loss of generality, we may assume that V=D. Then, we may choose r<1 sufficiently close to 1 so that Uis contained in the disk Drcentered at 0 with radius r.Weset Ur=f−1(Dr) so that f:Ur→Dris a ramified covering of degree d.Forris sufficiently close to 1, the boundary of Uris a union of R-analytic Jordan curves (canonically oriented by Ur) and fis holomorphic in a neighborhood of Ur. Observe that the fixed point of fare contained in Ur.Wemust show that f:Ur→Drhas dfixed points counting multiplicities. For any z∈∂Ur,wehave|z|<|f(z)|.Thus, by Rouch´e’s Theorem, f and Id −fhave the same number of zeros in Ur, counting multiplicities. Since f:Ur→Dris a ramified covering of degree d,0has dpre-images counted with multiplicities. Therefore, there are dfixed points. We must now show that fis repelling on average. The sum of residue fixed point indices is given by the integral 1 2iπ ∂Ur dz z−f(z). Virtually Repelling Fixed Points 203 We must prove that the real part of this integral is less than d/2. We can decompose it as follows: 1 2iπ ∂Ur dz z−f(z)=1 2iπ ∂Ur 1−f(z) z−f(z)dz +1 2iπ ∂Ur f(z) z−f(z)dz =1 2iπ ∂Ur f(z) f(z)dz +1 2iπ ∂Ur f(z) z−f(z) f(z) f(z)dz =1 2iπ ∂Ur z z−f(z) f(z) f(z)dz. Now, f:∂Ur→∂Dris orientation preserving, and thus, 1 2iπ f(z) f(z)dz is real and positive. Besides, when z∈∂Ur,wehave |f(z)/z|<1 and thus z z−f(z)<1 2. Therefore, ∂Ur z z−f(z)·1 2iπ f(z) f(z)dz=∂Ur z z−f(z)·1 2iπ f(z) f(z)dz <1 21 2iπ ∂Ur f(z) f(z)dz=d 2. This concludes the proof of Theorem 2. In fact, still under the assumption that Vis simply connected, one can better control the sum of residue fixed point indices. Let A⊂V be the connected component of V\Uwhich is not compactly contained in V. Observe that Ais an annulus. There exists a unique real ρ∈ ]0,1[ such that Ais conformally equivalent to the annulus D\[0,ρ]. The sum of residue fixed points indices of fis contained in the disk of diameter [dρ/(ρ−1), dρ/(ρ+ 1)]. The proof is very similar to the argument given above. Without loss of generality, we may assume that V=Dand 0 ∈U. Since the annulus Ais conformally equivalent to D\[0,ρ], it is known (see [A, Section 4]) that for any z∈U,we have |z|<ρ.Thus, when z∈∂Ur,z/(z−f(z)) belongs to the disk of diameter [ρ/(ρ−1),ρ/(ρ+ 1)]. Since 1 2iπ f(z) f(z)dz defines on ∂Ura