Some open problems in higher dimensional complex analysis and complex dynamics

Publ. Mat. 45 (2001), 529–547 SOME OPEN PROBLEMS IN HIGHER DIMENSIONAL COMPLEX ANALYSIS AND COMPLEX DYNAMICS John Erik Fornæss∗and Nessim Sibony Abstract We present a collection of problems in complex analysis and complex dynamics in several variables. Contents 1. Introduction 529 2. Complex dynamics problem list 530 2.1. Discrete dynamics 530 2.1.1. Fatou sets, stable sets 530 2.1.2. Julia sets, currents 532 2.2. Continuous dynamics 535 3. Several domplex variable problem list 536 3.1. ¯ ∂536 3.2. Levi problem 538 3.3. Holomorphic mappings 539 References 540 1. Introduction We present here a collection of problems in complex analysis and complex dynamics in several variables. The list contains some old questions which are well known and some new ones. There is no pretention to be exhaustive and the leading line was just that one of us was interested recently in these problems. The questions are of various nature, some would lead to a breakthrough and require new ideas, others are reasonably easy. We thank Bedford, Berndtsson, Burns, Henkin, 2000 Mathematics Subject Classification. 32D, 32F05, 32F20, 32H40, 32H50, 53C65, 58F05, 58F11, 58F18. Key words. Fatou set, biholomorphic map, Siegel domain, dimension of measures, symplectic maps, ¯ ∂-equation, Levi problem, holomorphic mappings. ∗The first author was supported by an NSF grant. 530 J. E. Fornæss, N. Sibony Ohsawa, Pinchuk, who have proposed some of the questions. We also thank Diederich, Ebenfeldt, Jonsson, Slapar and Winkelmann for comments and suggestions on the original version and Siu for providing the solution to one of the problems. 2. Complex dynamics problem list We divide the questions into discrete and continuous ones. The discrete questions are further divided in two sets depending on whether the stable or chaotic features are most dominant. 2.1. Discrete dynamics. 2.1.1. Fatou sets, stable sets. Question 2.1. Let Fbe the family of holomorphic endomorphisms of Pkof degree d≥2with infinitely many sinks. (We call the basin of attraction of an attracting periodic point a sink.) Can Fhave interior or have positive measure? When k=1it is a result of Fatou [CG] that there are finitely many sinks. (One needs a critical point in each basin.) In more variables, using the Newhouse phenomenon (persistence of homoclinic tangencies), Gavosto and Buzzard constructed infinitely many sinks. The question is how frequently this happens. See [Ga] and [Bu]. Question 2.2. Can maps on Pk,k≥2have wandering Fatou components? Recall that a Fatou component Ωis wandering for fif {fn(Ω)}n≥0 are two by two disjoint. When k=1, Sullivan, using the Ahlfors-Bers Theorem proved that there are no wandering components [CG]. The tool used is not available in several variables. Maybe the first step is to find a new proof in one variable. A domain U⊂Ckis a Fatou-Bieberbach domain if it is biholomorphic to Ckbut U=Ck. Question 2.3. Can Fatou-Bieberbach domains for H´enon maps have C∞boundary? Recall that a H´enon map in C2is a biholomorphism fof the following form: f(z,w)=(p(z)+aw, bz),ab=0, with pa polynomial of degree d≥2. Assuming that 0is an attracting fixed point it is known that Ω:={q;fn(q)→p}is biholomorphic to C2, but Ω=C2. The question is about the smoothness of ∂Ω. It is known that there are domains U⊂C2,U=C2,Ubiholomorphic to C2and Uhas smooth boundary [St]. Problems in Complex Analysis and Dynamics 531 Question 2.4. Let fbe a generalized H´enon map on C2.Let K±={(z,w); {f±n}n≥0is a bounded sequence}, J±=∂K±. There is a unique positive closed current T±of norm 1 supported on K±[FS10]. The measure µ=T+∧T−has been studied intensely [BLS]. The nonwandering set Ω(f)for fis contained in K:= K+∩K−. What are the relations between Ω(f),J:= J+∩J−and J∗=Sµ, the support of µ? When is fhyperbolic on Ω(f)? There are a few cases known, perturbations of (zd+c, 0),[FS3],[HO], more recently Hruska [H]has computer assisted examples of hyperbolic maps. Question 2.5. Let Mkbe a complex manifold. Assume that M= ∪φn(B),φn:B→M,1−1and φn+1(B)⊃φn(B),Bis the unit ball. Assume that the Kobayashi metric KM≡0.IsM=Ck? (False when k≥3, open when k=2[FS11].) An example of this situation is a question raised by Bedford: Let F:Cn→Cnbe a biholomorphism, Ka hyperbolic saddle. Assume that the stable dimension of Kis k.Forp∈K,isWs(p)biholomorphic to Ck? This is true for periodic saddle points. It is easy to show that the Kobayashi metric vanishes on all stable leaves. See [JV]for results on generic stable manifolds. Question 2.6. Is any “long” C2biholomorphic to C2?(A complex manifold is a long C2if it is a union of proper subsets which are biholomorphic to C2.) Question 2.7 (M. Herman).Symplectic question: H=(f(z)−w,z), fentire. Find fsuch that fhas a Siegel domain at 0,f(0)=0. The problem is that the eigenvalues of H(0) are e±iθ so we have resonances. Formally one gets infinitely many compatibility conditions. Recall that a Siegel domain is a domain where a subsequence of iterates converges to the identity. Question 2.8. Can a Siegel domain for a H´enon map or, more generally, a biholomorphism of C2have strongly pseudoconvex boundary? In one variable, R. P´erez-Marco has constructed Siegel discs with smooth boundary [PM]. Question 2.9. Classify periodic Fatou components for an endomorphism of Pk[FS6].Letf:Pk→Pkbe an endomorphism. Assume that U is a Fatou component such that f(U)=U. Assume that (fn)(z)→∂U. Does there exist a parabolic fixed point on ∂U?See[JL]. When k=1 this was done by Fatou [CG]. 532 J. E. Fornæss, N. Sibony Question 2.10. Classify those Reinhardt domains which are biholomorphic to Siegel domains, associated to an automorphism of Ckor to an endomorphism of Pk. Question 2.11. Suppose that the boundary of a Fatou component of aH´enon map has a smooth piece. What can be said about the whole boundary? In one variable the whole boundary must be smooth. Question 2.12 (The kicked rotor; infinite dimensional complex dynamics).The kicked rotor describes a particle on a circular orbit subject to a periodic kick. In quantum mechanics it describes an elementary particle in an atom subject to a periodically pulsed electric field. Classical description: If angular momentum is p, mass is m,Kis the kick strength, Tis the time between kicks, then R(θ, p)=(θ+pT m,p+ Ksin(θ+pT m)) describes the change in coordinates from just after one kick till just after the next kick. (This is also called the Standard Map.) Prove that if k=Km T>1then almost all orbits are unbounded. Quantum description: If ψ=n∈Zcneinθ is an L2function on [0,2π] describing the state of the particle after one kick, then after the next kick, the state is R(ψ)=e−iKcos θ ne−in2T 2mcneinθ. Show that the dynamics localizes. More precisely, if Rk(ψ)=nck neinθ then for some C=C(ψ)we have |n|≤C|ck n|2>1/2for all k. It is necessary to avoid resonances [CCIF]. Computer experiments support this statement [Gr]. See [F2],[We]for rigorous results for a simplified model. Question 2.13. Define chaos in the setting of infinite dimensional complex dynamics. See [F2]. Question 2.14. Suppose that fis an endomorphism of Pkor an automorphism of Ckwith an attracting basin Ωfor an attracting fixed point p, f(p)=p.Letδ>0.Forq∈Ω,dist(q,∂Ω) >δ,dist(q,p)>δ, i.e. q∈Ω∗ δ. Find N(δ)so that no orbit {q,f(q),...,fN(q)}⊂Ω∗ δ. Example (z2+&w, z)with p=0. The question here is to get explicit estimates of N(δ)in terms of f,pand δ. The question is to have concrete estimates on how many iterates you need to get close to the fixed point if you start near the boundary of the basin. 2.1.2. Julia sets, currents. Question 2.15. Let fbe a holomorphic selfmap of Pk.LetΩ(f)be the nonwandering set. Recall that a point is nonwandering if for every neighborhood Uof p, there is an n>1such that fn(U)∩U=∅. The question is to describe Ω(f). Problems in Complex Analysis and Dynamics 533 Define a Siegel set as a closed analytic set Xin an open set V⊂ Pksuch that there is a subsequence ni→∞with fni |X→Id|X.Is Ω(f)the closure of the periodic points together with Siegel analytic sets? See [FS3],[FS6],[BS2]. Can there be a counterexample for entire maps on C2?(Probably yes.) Question 2.16. Let fbe a holomorphic selfmap of a complex manifold M. Is the closure of the repelling periodic orbits open in Ω(f)? Can there be a counterexample for entire maps in Cn?Could there be a sequence of oscillating orbits converging to a repelling point? Can a sequence of saddles converge to a repelling point for a holomorphic selfmap on C2? (This cannot happen for biholomorphisms because for the inverse map the repelling point becomes attracting.) We say that the orbit of a point pis oscillating if some subsequence is uniformly bounded and some other subsequence converges to infinity. For construction of oscillating domains see [FS5]. On dynamics of transcendental maps see [FS7],[FS9]. Question 2.17. Let fbe a polynomial map on Ckof topological degree dt, which extends to a holomorphic map on Pk.Letωbe the K¨ahler form on Pk. It is known that (fn)∗ωk dnk t →µ where µis a mixing probability measure [FS1]. The measure µis the unique measure of maximal entropy [BD2]and repelling periodic points are dense in Sµ, the support of µ[BD1].SoSµ⊂Ω(f). Is the support of µopen in Ω(f)? It is the case when Sµis hyperbolic. Recall that the Hausdorff dimension of a probability measure νis the minimal Hausdorff dimension of a Borel set with ν(B)=1. For a polynomial map in C,µcoincides with the harmonic measure with respect to ∞of the compact KP:= {z;Pn(z)is bounded}. It follows from the work of Makarov [Ma], Bishop-Jones [BJ], Wolff [W] that the Hausdorff dimension of µis 1. How does this result extend to polynomial maps on Ckthat are holomorphic on Pk? What is the Hausdorff dimension of µfor an endomorphism of Pk? A first case is to study small perturbations (z2+&w, w2+δz). When is µabsolutely continuous with respect to Lebesgue measure. Question 2.18. The Fatou set of an endomorphism fof Pkis the maximal open set where the sequence fnis locally equicontinuous. The Julia set is the complement. It coincides with the support of the positive 534 J. E. Fornæss, N. Sibony closed current T= lim (fn)∗ω dn, here dis the algebraic degree of f[FS1], [FS4]. The Julia set Jof an endomorphism on Pkis not in general contained in the nonwandering set (contrary to what happens when k=1). Let J0:= Ω(fJ). What can be said about the Hausdorff dimension of J,J0,J\J 0?Describe J0\Supp(µ). When k=1,J⊂J0. Question 2.19. Given an endomorphism of Pk, when is the current T extremal? This is true for maps of the form [P1:··· :Pk:td][BeSi]. Here the polynomials Pjare homogeneous polynomials of degree din z and do not contain a term in t. Jonsson has observed that for Ueda type examples the Green current is not extremal, for example, this is the case for the map [z2:w2−2zt :t2]. Question 2.20. We say that an endomorphism of Pkis critically finite if each irreducible component of the critical hypersurface is a preperiodic set. Find families of endomorphisms of P2of critically finite maps [FS12],[J]. Question 2.21. For fan endomorphism of Pk, when is Supp(µ)=Pk? For how many maps? Say, find the Hausdorff dimension of this set of maps. When is µabsolutely continuous with respect to Lebesgue measure. When k=1,Sµ=P1if and only if the Julia set of fis P1. If one considers rational maps of degree d≥2on P1, this happens for a set of positive Lebesgue measure [R]. Question 2.22. Let fbe a holomorphic endomorphism of Pk,k>1. A closed set Ais attracting if there is an open set U(A)such that f(U(A)) ⊂⊂ U(A),∩fn(U)=A.Ais an attractor if it is attracting and has a dense orbit. (Sometimes one assumes instead that it is chain transitive, i.e. δ-pseudo-orbits are dense.) The attractor is non-trivial if it is not a periodic orbit or the whole space. Show that non trivial attractors are robust in P2, i.e. for the endomorphisms (fc)on P2, find a set of positive measure in the c-space with a nontrivial attractor. Give estimates for the Hausdorff dimension of A. For examples of non trivial attractors on P2see [JW],[FS8]. There are some estimates on Hausdorff dimension of attractors in [FS8]. Question 2.23. Does there exist a H´enon map f=(g,h),fn=(gn,h n) with the following property: There is a p=(x, y)∈Sµsuch that for all 0<&<<1and every q∈Sµ, there is an integer nso that gn(p)− gn(q)<&?This question arises naturally in the study of attractors: A collision-attractor absorbs all points whose orbits are closer than some radius for some iterate. If one considers xas a space variable, and y Problems in Complex Analysis and Dynamics 535 as a momentum variable, one measures distance using the first variable only. If one replaces the inequality by fn(p)−fn(q)<&, there is no such p, (see [BF]). Question 2.24. Let f(z,w)=(eiθz,e−iθw)+ higher order terms be a generalized H´enon map. Give diophantine conditions on θfor which f has two Siegel discs, D1,D 2⊂J+∩J−through the origin and tangent to the axes. Recall that generalized H´enon maps are finite compositions of H´enon maps. They are the dynamically interesting polynomial biholomorphisms of C2. Question 2.25. Consider SR, the class of holomorphic symplectomorphisms of C2kpreserving R2k.f:C2k→C2k,f∗ω=ω, where ω= k j=1 dzj∧dwjand f(R2k)=R2k,zj=xj+ix j,wj=yj+iy j with Whitney fine topology, SRis a Baire space. For f∈S R, let KR f:= {(x, y)∈R2k;{fn(x, y)}nis bounded}. Prove that the set S R of f∈S Rsuch that KR fis of empty interior in R2kis a Gδdense set [FS7],[FS9]. Question 2.26. Let f(z,w)=(eiθz,eiψw)+ higher order terms. Study the dynamics near the origin. Hakim-Abate-Weickert have studied the case where fis tangent to the identity [Ha1],[Ha2][Ab],[We]. Question 2.27 (D. Burns).Suppose that f:XC→XCis an endomorphism on a projective variety over C. Assume that fhas “large” dynamics. For example fhas positive entropy or a “large” non wandering set. Consider now X(k)as a variety over a number field ksuch that [k:Q]<∞. Does largeness of the dynamics imply largeness of the set of rational points in X(k)?For example, are rational points Zariski dense in XC? (J. B. Bost). The existence of mappings with large dynamics should imply arithmetic properties of X(k). Question 2.28. Let Kbe a hyperbolic solenoid with stable dimension 2. How to get a stable current? Are the stable leaves biholomorphic to C2? For endomorphisms on Pksee [FS4], [FS2],[FS8], [FS11], [F1], [S1], [BD1], [BD2], [Ga], [DS]. 2.2. Continuous dynamics. Question 2.29. Is there a compact set K⊂P2which is laminated by smooth holomorphic curves, except a compact curve? There are such compacts in P3. See survey by Ghys [Gh]. There is no closed (1,1) current directed by the lamination in P2. (I.e. if locally the current is of the form dµθ[Vθ], then the lamination is a compact curve.) See [HM]. 536 J. E. Fornæss, N. Sibony The origin of the question seems to be related to a Poincar´e-Bendixson Theorem for holomorphic foliations on P2,[CLS]. More precisely, let Fbe a holomorphic foliation on P2. The singularity set of F,Sing Fis never empty. The question is whether the closure of any leaf Lintersects Sing F. If not Lwill be laminated by smooth holomorphic curves [CLS]. Question 2.30. Let PNconsist of the holomorphic polynomials of degree at most Nin C2k,k≥2,N≥2. Show that for almost every PNand almost every point z∈C2kthe orbit of the Hamiltonian vector field XPNis unbounded. Recall that when his an entire map in C2k, Xh:= −∂h ∂w1 ,...,−∂h ∂wk ,∂h ∂z1 ,..., ∂h ∂zk. [FS7],[FS9]contain results on Hamiltonian vector fields and dynamical symplectomorphisms. (When k=1,see[Du].) 3. Several complex variable problem list We divide the questions into three sets, depending whether they fit most naturally together with ¯ ∂, the Levi Problem or Holomorphic Mappings 3.1. ¯ ∂. Question 3.1 (Henkin).Let Xbe a normal analytic set of pure dimension pin the unit ball B⊂Cn.Letf∈C∞ (0,1)(B). Assume ¯ ∂Id∗ |Reg(X)f= 0. Does there exist u∈C∞ (0,0)(B)such that ¯ ∂Id∗ Reg(X)u=Id∗ |Reg(X)f? Here IdReg(X)is the inclusion map from Reg(X)to Cn.See[HeP],[M], [AG1],[AG2]. Question 3.2. Let Ωbe a weakly pseudoconvex domain with smooth boundary in Pn,n≥2.Letf∈C ∞ (0,1)(Ω) be a ¯ ∂-closed (0,1) form. Does there exist a smooth solution u∈C ∞(Ω) to the equation ¯ ∂u =f? Question 3.3 (Lempert-Henkin).Let DN={z∈CN;N j=1 |zj|<1}. Prove (or find a counterexample) that for f∈C (0,1)(DN)with ¯ ∂f =0 there is u∈C(DN)such that ¯ ∂u =fand u∞≤Cf∞. The key point here is that Cshould be independent of N. One can ask the same question for D2 N={z∈CN;N j=1 |zj|2<1}[L]. Problems in Complex Analysis and Dynamics 537 Question 3.4 (Berndtsson).Let φbe a bounded strictly plurisubharmonic function in the unit ball B⊂Cn.LetΩ=i∂ ¯ ∂φ, the K¨ahler form associated to φ.Letfbe a (0,1) form on Bsuch that |f|2 Ω+|∂f|Ω≤C. The subscript means that the norm for fand ∂f are measured with respect to Ω. Does the equation ¯ ∂u =fhave a bounded solution in B? The question is related to the Corona theorem in several variables [Be]. Question 3.5 (Edited by Sophia Vassiliadou).Let Xbe a closed analytic set of pure dimension pin Cn+pwith singularities. Let Reg(X)= X\Sing(X)denote the set of smooth points of X. Give Reg(X)the metric induced by the imbedding Reg(X)8→Cn+p. (a) Let f∈L2 (0,q)(Reg(X)),1≤q≤p,¯ ∂f =0in the weak sense in L2 0,q+1(Reg(X)). Find the obstructions to solving ¯ ∂u =fin the weak sense in L2 0,q(Reg(X)). Recall that ¯ ∂u =fin the weak sense in L2 0,q(Reg(X)) if and only if u∈L2 0,q−1(Reg(X)) and for all ψ∈C ∞ p,p−q(Reg(X)), compactly supported in Reg(X)we have Reg(X) u∧¯ ∂ψ =(−1)qReg(X) f∧ψ. (b) Let h∈L2 (α,β)(Reg(X)). We say that hbelongs to the domain of ¯ ∂with Dirichlet boundary conditions —h∈Dom ¯ ∂D— if and only if there exist hn∈C ∞ α,β(Reg(X)), compactly supported in Reg(X) and g∈L2 α,β+1(Reg(X)) such that hn→hin L2 α,β(Reg(X)) and ¯ ∂hn→gin L2 α,β+1(Reg(X)). In that case we write ¯ ∂Dh=: g. Let f∈L2 (p,q)(Reg(X)),1≤q≤psuch that ¯ ∂Df=0. Find the obstructions to solving ¯ ∂Du=f, (i.e. obstructions to obtaining un∈C ∞ (p,q−1)(Reg(X)), compactly supported in Reg(X)such that un→uin L2 (p,q−1)(Reg(X)) and ¯ ∂un→fin L2 (p,q)(Reg(X))). In [PS], (a) and (b) are solved when Xis a projective surface with isolated singularity. They also computed obstructions to solving ¯ ∂weakly for ¯ ∂closed (p, q)forms when Xis a projective variety of dimension p, as well as obstructions to solving ¯ ∂Dfor (0,q),¯ ∂D-closed forms again when Xis a projective variety of dimension p. See [BeSi]for an alternative point of view: Solving ¯ ∂on positive currents (especially positive, closed currents of bidegree (1,1)). See [DFV]for obstructions to solving ¯ ∂weakly for (0,1) forms near 2-dimensional isolated singularities in Cn,n≥3. 544 J. E. Fornæss, N. Sibony [FS11] J. E. Fornæss and N. Sibony, Increasing sequences of complex manifolds, Math. Ann. 255(3) (1981), 351–360. [FS12] J. E. Fornæss and N. 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