Publ. Ma . 45 (2001), 529–547
SOME OPEN PROBLEMS IN HIGHER DIMENSIONAL
COMPLEX ANALYSIS AND COMPLEX DYNAMICS
John E ik Fo næss∗and Nessim Sibony
Abs ac
We p esen a collec ion o p oblems in complex analysis and com-
plex dynamics in se e al a iables.
Con en s
1. In oduc ion 529
2. Complex dynamics p oblem lis 530
2.1. Disc e e dynamics 530
2.1.1. Fa ou se s, s able se s 530
2.1.2. Julia se s, cu en s 532
2.2. Con inuous dynamics 535
3. Se e al domplex a iable p oblem lis 536
3.1. ¯
∂536
3.2. Le i p oblem 538
3.3. Holomo phic mappings 539
Re e ences 540
1. In oduc ion
We p esen he e a collec ion o p oblems in complex analysis and
complex dynamics in se e al a iables. The lis con ains some old ques-
ions which a e well known and some new ones. The e is no p e en-
ion o be exhaus i e and he leading line was jus ha one o us was
in e es ed ecen ly in hese p oblems. The ques ions a e o a ious na-
u e, some would lead o a b eak h ough and equi e new ideas, o he s
a e easonably easy. We hank Bed o d, Be nd sson, Bu ns, Henkin,
2000 Ma hema ics Subjec Classifica ion. 32D, 32F05, 32F20, 32H40, 32H50, 53C65,
58F05, 58F11, 58F18.
Key wo ds. Fa ou se , biholomo phic map, Siegel domain, dimension o measu es,
symplec ic maps, ¯
∂-equa ion, Le i p oblem, holomo phic mappings.
∗The fi s au ho was suppo ed by an NSF g an .
530 J. E. Fo næss, N. Sibony
Ohsawa, Pinchuk, who ha e p oposed some o he ques ions. We also
hank Diede ich, Eben eld , Jonsson, Slapa and Winkelmann o com-
men s and sugges ions on he o iginal e sion and Siu o p o iding he
solu ion o one o he p oblems.
2. Complex dynamics p oblem lis
We di ide he ques ions in o disc e e and con inuous ones. The dis-
c e e ques ions a e u he di ided in wo se s depending on whe he he
s able o chao ic ea u es a e mos dominan .
2.1. Disc e e dynamics.
2.1.1. Fa ou se s, s able se s.
Ques ion 2.1. Le Fbe he amily o holomo phic endomo phisms o
Pko deg ee d≥2wi h infini ely many sinks. (We call he basin o
a ac ion o an a ac ing pe iodic poin a sink.) Can Fha e in e io
o ha e posi i e measu e? When k=1i is a esul o Fa ou [CG]
ha he e a e fini ely many sinks. (One needs a c i ical poin in each
basin.) In mo e a iables, using he Newhouse phenomenon (pe sis ence
o homoclinic angencies), Ga os o and Buzza d cons uc ed infini ely
many sinks. The ques ion is how equen ly his happens. See [Ga]
and [Bu].
Ques ion 2.2. Can maps on Pk,k≥2ha e wande ing Fa ou compo-
nen s? Recall ha a Fa ou componen Ωis wande ing o i { n(Ω)}n≥0
a e wo by wo disjoin . When k=1, Sulli an, using he Ahl o s-Be s
Theo em p o ed ha he e a e no wande ing componen s [CG]. The ool
used is no a ailable in se e al a iables. Maybe he fi s s ep is o find
a new p oo in one a iable.
A domain U⊂Ckis a Fa ou-Biebe bach domain i i is biholomo phic
o Ckbu U=Ck.
Ques ion 2.3. Can Fa ou-Biebe bach domains o H´enon maps ha e
C∞bounda y? Recall ha a H´enon map in C2is a biholomo phism o
he ollowing o m:
(z,w)=(p(z)+aw, bz),ab=0,
wi h pa polynomial o deg ee d≥2. Assuming ha 0is an a ac ing
fixed poin i is known ha Ω:={q; n(q)→p}is biholomo phic o C2,
bu Ω=C2. The ques ion is abou he smoo hness o ∂Ω.
I is known ha he e a e domains U⊂C2,U=C2,Ubiholomo phic
o C2and Uhas smoo h bounda y [S ].
P oblems in Complex Analysis and Dynamics 531
Ques ion 2.4. Le be a gene alized H´enon map on C2.Le
K±={(z,w); { ±n}n≥0is a bounded sequence},
J±=∂K±. The e is a unique posi i e closed cu en T±o no m 1
suppo ed on K±[FS10]. The measu e µ=T+∧T−has been s udied
in ensely [BLS]. The nonwande ing se Ω( ) o is con ained in K:=
K+∩K−. Wha a e he ela ions be ween Ω( ),J:= J+∩J−and
J∗=Sµ, he suppo o µ? When is hype bolic on Ω( )? The e a e a
ew cases known, pe u ba ions o (zd+c, 0),[FS3],[HO], mo e ecen ly
H uska [H]has compu e assis ed examples o hype bolic maps.
Ques ion 2.5. Le Mkbe a complex mani old. Assume ha M=
∪φn(B),φn:B→M,1−1and φn+1(B)⊃φn(B),Bis he uni
ball. Assume ha he Kobayashi me ic KM≡0.IsM=Ck? (False
when k≥3, open when k=2[FS11].)
An example o his si ua ion is a ques ion aised by Bed o d: Le
F:Cn→Cnbe a biholomo phism, Ka hype bolic saddle. Assume
ha he s able dimension o Kis k.Fo p∈K,isWs(p)biholomo phic
o Ck? This is ue o pe iodic saddle poin s. I is easy o show ha
he Kobayashi me ic anishes on all s able lea es. See [JV] o esul s
on gene ic s able mani olds.
Ques ion 2.6. Is any “long” C2biholomo phic o C2?(A complex man-
i old is a long C2i i is a union o p ope subse s which a e biholomo -
phic o C2.)
Ques ion 2.7 (M. He man).Symplec ic ques ion: H=( (z)−w,z),
en i e. Find such ha has a Siegel domain a 0, (0)=0. The
p oblem is ha he eigen alues o H(0) a e e±iθ so we ha e esonances.
Fo mally one ge s infini ely many compa ibili y condi ions. Recall ha a
Siegel domain is a domain whe e a subsequence o i e a es con e ges o
he iden i y.
Ques ion 2.8. Can a Siegel domain o a H´enon map o , mo e gene -
ally, a biholomo phism o C2ha e s ongly pseudocon ex bounda y? In
one a iable, R. P´e ez-Ma co has cons uc ed Siegel discs wi h smoo h
bounda y [PM].
Ques ion 2.9. Classi y pe iodic Fa ou componen s o an endomo ph-
ism o Pk[FS6].Le :Pk→Pkbe an endomo phism. Assume ha U
is a Fa ou componen such ha (U)=U. Assume ha ( n)(z)→∂U.
Does he e exis a pa abolic fixed poin on ∂U?See[JL]. When k=1
his was done by Fa ou [CG].
532 J. E. Fo næss, N. Sibony
Ques ion 2.10. Classi y hose Reinha d domains which a e biholomo -
phic o Siegel domains, associa ed o an au omo phism o Cko o an
endomo phism o Pk.
Ques ion 2.11. Suppose ha he bounda y o a Fa ou componen o
aH´enon map has a smoo h piece. Wha can be said abou he whole
bounda y? In one a iable he whole bounda y mus be smoo h.
Ques ion 2.12 (The kicked o o ; infini e dimensional complex dynam-
ics).The kicked o o desc ibes a pa icle on a ci cula o bi subjec o a
pe iodic kick. In quan um mechanics i desc ibes an elemen a y pa icle
in an a om subjec o a pe iodically pulsed elec ic field.
Classical desc ip ion: I angula momen um is p, mass is m,Kis he
kick s eng h, Tis he ime be ween kicks, hen R(θ, p)=(θ+pT
m,p+
Ksin(θ+pT
m)) desc ibes he change in coo dina es om jus a e one
kick ill jus a e he nex kick. (This is also called he S anda d Map.)
P o e ha i k=Km
T>1 hen almos all o bi s a e unbounded.
Quan um desc ip ion: I ψ=n∈Zcneinθ is an L2 unc ion on [0,2π]
desc ibing he s a e o he pa icle a e one kick, hen a e he nex
kick, he s a e is R(ψ)=e−iKcos θ
ne−in2T
2mcneinθ. Show ha he
dynamics localizes. Mo e p ecisely, i Rk(ψ)=nck
neinθ hen o some
C=C(ψ)we ha e |n|≤C|ck
n|2>1/2 o all k. I is necessa y o a oid
esonances [CCIF]. Compu e expe imen s suppo his s a emen [G ].
See [F2],[We] o igo ous esul s o a simplified model.
Ques ion 2.13. Define chaos in he se ing o infini e dimensional com-
plex dynamics. See [F2].
Ques ion 2.14. Suppose ha is an endomo phism o Pko an au o-
mo phism o Ckwi h an a ac ing basin Ω o an a ac ing fixed poin p,
(p)=p.Le δ>0.Fo q∈Ω,dis (q,∂Ω) >δ,dis (q,p)>δ, i.e.
q∈Ω∗
δ. Find N(δ)so ha no o bi {q, (q),..., N(q)}⊂Ω∗
δ. Exam-
ple (z2+&w, z)wi h p=0. The ques ion he e is o ge explici es ima es
o N(δ)in e ms o ,pand δ. The ques ion is o ha e conc e e es i-
ma es on how many i e a es you need o ge close o he fixed poin i
you s a nea he bounda y o he basin.
2.1.2. Julia se s, cu en s.
Ques ion 2.15. Le be a holomo phic sel map o Pk.Le Ω( )be
he nonwande ing se . Recall ha a poin is nonwande ing i o e e y
neighbo hood Uo p, he e is an n>1such ha n(U)∩U=∅. The
ques ion is o desc ibe Ω( ).
P oblems in Complex Analysis and Dynamics 533
Define a Siegel se as a closed analy ic se Xin an open se V⊂
Pksuch ha he e is a subsequence ni→∞wi h ni
|X→Id|X.Is
Ω( ) he closu e o he pe iodic poin s oge he wi h Siegel analy ic se s?
See [FS3],[FS6],[BS2]. Can he e be a coun e example o en i e maps
on C2?(P obably yes.)
Ques ion 2.16. Le be a holomo phic sel map o a complex mani-
old M. Is he closu e o he epelling pe iodic o bi s open in Ω( )?
Can he e be a coun e example o en i e maps in Cn?Could he e be
a sequence o oscilla ing o bi s con e ging o a epelling poin ? Can a
sequence o saddles con e ge o a epelling poin o a holomo phic sel -
map on C2? (This canno happen o biholomo phisms because o he
in e se map he epelling poin becomes a ac ing.) We say ha he o -
bi o a poin pis oscilla ing i some subsequence is uni o mly bounded
and some o he subsequence con e ges o infini y. Fo cons uc ion o
oscilla ing domains see [FS5]. On dynamics o anscenden al maps
see [FS7],[FS9].
Ques ion 2.17. Le be a polynomial map on Cko opological de-
g ee d , which ex ends o a holomo phic map on Pk.Le ωbe he K¨ahle
o m on Pk. I is known ha
( n)∗ωk
dnk
→µ
whe e µis a mixing p obabili y measu e [FS1]. The measu e µis he
unique measu e o maximal en opy [BD2]and epelling pe iodic poin s
a e dense in Sµ, he suppo o µ[BD1].SoSµ⊂Ω( ). Is he suppo
o µopen in Ω( )? I is he case when Sµis hype bolic.
Recall ha he Hausdo ff dimension o a p obabili y measu e νis he
minimal Hausdo ff dimension o a Bo el se wi h ν(B)=1.
Fo a polynomial map in C,µcoincides wi h he ha monic measu e
wi h espec o ∞o he compac KP:= {z;Pn(z)is bounded}. I ol-
lows om he wo k o Maka o [Ma], Bishop-Jones [BJ], Wolff [W]
ha he Hausdo ff dimension o µis 1. How does his esul ex end o
polynomial maps on Ck ha a e holomo phic on Pk?
Wha is he Hausdo ff dimension o µ o an endomo phism o Pk?
A fi s case is o s udy small pe u ba ions (z2+&w, w2+δz). When is
µabsolu ely con inuous wi h espec o Lebesgue measu e.
Ques ion 2.18. The Fa ou se o an endomo phism o Pkis he max-
imal open se whe e he sequence nis locally equicon inuous. The Ju-
lia se is he complemen . I coincides wi h he suppo o he posi i e
534 J. E. Fo næss, N. Sibony
closed cu en T= lim ( n)∗ω
dn, he e dis he algeb aic deg ee o [FS1],
[FS4]. The Julia se Jo an endomo phism on Pkis no in gene al con-
ained in he nonwande ing se (con a y o wha happens when k=1).
Le J0:= Ω( J). Wha can be said abou he Hausdo ff dimension o
J,J0,J J
0?Desc ibe J0 Supp(µ). When k=1,J⊂J0.
Ques ion 2.19. Gi en an endomo phism o Pk, when is he cu en T
ex emal? This is ue o maps o he o m [P1:··· :Pk: d][BeSi].
He e he polynomials Pja e homogeneous polynomials o deg ee din z
and do no con ain a e m in . Jonsson has obse ed ha o Ueda ype
examples he G een cu en is no ex emal, o example, his is he case
o he map [z2:w2−2z : 2].
Ques ion 2.20. We say ha an endomo phism o Pkis c i ically fi-
ni e i each i educible componen o he c i ical hype su ace is a p epe-
iodic se . Find amilies o endomo phisms o P2o c i ically fini e
maps [FS12],[J].
Ques ion 2.21. Fo an endomo phism o Pk, when is Supp(µ)=Pk?
Fo how many maps? Say, find he Hausdo ff dimension o his se o
maps. When is µabsolu ely con inuous wi h espec o Lebesgue measu e.
When k=1,Sµ=P1i and only i he Julia se o is P1. I one
conside s a ional maps o deg ee d≥2on P1, his happens o a se o
posi i e Lebesgue measu e [R].
Ques ion 2.22. Le be a holomo phic endomo phism o Pk,k>1.
A closed se Ais a ac ing i he e is an open se U(A)such ha
(U(A)) ⊂⊂ U(A),∩ n(U)=A.Ais an a ac o i i is a ac ing
and has a dense o bi . (Some imes one assumes ins ead ha i is chain
ansi i e, i.e. δ-pseudo-o bi s a e dense.) The a ac o is non- i ial i
i is no a pe iodic o bi o he whole space.
Show ha non i ial a ac o s a e obus in P2, i.e. o he endo-
mo phisms ( c)on P2, find a se o posi i e measu e in he c-space wi h
a non i ial a ac o .
Gi e es ima es o he Hausdo ff dimension o A. Fo examples o
non i ial a ac o s on P2see [JW],[FS8]. The e a e some es ima es
on Hausdo ff dimension o a ac o s in [FS8].
Ques ion 2.23. Does he e exis a H´enon map =(g,h), n=(gn,h
n)
wi h he ollowing p ope y: The e is a p=(x, y)∈Sµsuch ha o all
0<&<<1and e e y q∈Sµ, he e is an in ege nso ha gn(p)−
gn(q)<&?This ques ion a ises na u ally in he s udy o a ac o s: A
collision-a ac o abso bs all poin s whose o bi s a e close han some
adius o some i e a e. I one conside s xas a space a iable, and y
P oblems in Complex Analysis and Dynamics 535
as a momen um a iable, one measu es dis ance using he fi s a iable
only. I one eplaces he inequali y by n(p)− n(q)<&, he e is no
such p, (see [BF]).
Ques ion 2.24. Le (z,w)=(eiθz,e−iθw)+ highe o de e ms be a
gene alized H´enon map. Gi e diophan ine condi ions on θ o which
has wo Siegel discs, D1,D
2⊂J+∩J− h ough he o igin and angen
o he axes. Recall ha gene alized H´enon maps a e fini e composi ions
o H´enon maps. They a e he dynamically in e es ing polynomial biholo-
mo phisms o C2.
Ques ion 2.25. Conside SR, he class o holomo phic symplec omo -
phisms o C2kp ese ing R2k. :C2k→C2k, ∗ω=ω, whe e ω=
k
j=1 dzj∧dwjand (R2k)=R2k,zj=xj+ix
j,wj=yj+iy
j
wi h Whi ney fine opology, SRis a Bai e space. Fo ∈S
R, le
KR
:= {(x, y)∈R2k;{ n(x, y)}nis bounded}. P o e ha he se S
R
o ∈S
Rsuch ha KR
is o emp y in e io in R2kis a Gδdense
se [FS7],[FS9].
Ques ion 2.26. Le (z,w)=(eiθz,eiψw)+ highe o de e ms. S udy
he dynamics nea he o igin. Hakim-Aba e-Weicke ha e s udied he
case whe e is angen o he iden i y [Ha1],[Ha2][Ab],[We].
Ques ion 2.27 (D. Bu ns).Suppose ha :XC→XCis an endomo -
phism on a p ojec i e a ie y o e C. Assume ha has “la ge” dy-
namics. Fo example has posi i e en opy o a “la ge” non wande ing
se . Conside now X(k)as a a ie y o e a numbe field ksuch ha
[k:Q]<∞. Does la geness o he dynamics imply la geness o he se
o a ional poin s in X(k)?Fo example, a e a ional poin s Za iski dense
in XC? (J. B. Bos ). The exis ence o mappings wi h la ge dynamics
should imply a i hme ic p ope ies o X(k).
Ques ion 2.28. Le Kbe a hype bolic solenoid wi h s able dimension 2.
How o ge a s able cu en ? A e he s able lea es biholomo phic o C2?
Fo endomo phisms on Pksee [FS4], [FS2],[FS8], [FS11], [F1], [S1],
[BD1], [BD2], [Ga], [DS].
2.2. Con inuous dynamics.
Ques ion 2.29. Is he e a compac se K⊂P2which is lamina ed by
smoo h holomo phic cu es, excep a compac cu e? The e a e such
compac s in P3. See su ey by Ghys [Gh]. The e is no closed (1,1)
cu en di ec ed by he lamina ion in P2. (I.e. i locally he cu en is o
he o m dµθ[Vθ], hen he lamina ion is a compac cu e.) See [HM].
536 J. E. Fo næss, N. Sibony
The o igin o he ques ion seems o be ela ed o a Poinca ´e-Bendixson
Theo em o holomo phic olia ions on P2,[CLS]. Mo e p ecisely, le
Fbe a holomo phic olia ion on P2. The singula i y se o F,Sing Fis
ne e emp y. The ques ion is whe he he closu e o any lea Lin e sec s
Sing F. I no Lwill be lamina ed by smoo h holomo phic cu es [CLS].
Ques ion 2.30. Le PNconsis o he holomo phic polynomials o de-
g ee a mos Nin C2k,k≥2,N≥2. Show ha o almos e e y
PNand almos e e y poin z∈C2k he o bi o he Hamil onian ec o
field XPNis unbounded. Recall ha when his an en i e map in C2k,
Xh:= −∂h
∂w1
,...,−∂h
∂wk
,∂h
∂z1
,..., ∂h
∂zk.
[FS7],[FS9]con ain esul s on Hamil onian ec o fields and dynamical
symplec omo phisms. (When k=1,see[Du].)
3. Se e al complex a iable p oblem lis
We di ide he ques ions in o h ee se s, depending whe he hey fi
mos na u ally oge he wi h ¯
∂, he Le i P oblem o Holomo phic Map-
pings
3.1. ¯
∂.
Ques ion 3.1 (Henkin).Le Xbe a no mal analy ic se o pu e dimen-
sion pin he uni ball B⊂Cn.Le ∈C∞
(0,1)(B). Assume ¯
∂Id∗
|Reg(X) =
0. Does he e exis u∈C∞
(0,0)(B)such ha ¯
∂Id∗
Reg(X)u=Id∗
|Reg(X) ?
He e IdReg(X)is he inclusion map om Reg(X) o Cn.See[HeP],[M],
[AG1],[AG2].
Ques ion 3.2. Le Ωbe a weakly pseudocon ex domain wi h smoo h
bounda y in Pn,n≥2.Le ∈C
∞
(0,1)(Ω) be a ¯
∂-closed (0,1) o m. Does
he e exis a smoo h solu ion u∈C
∞(Ω) o he equa ion ¯
∂u = ?
Ques ion 3.3 (Lempe -Henkin).Le DN={z∈CN;N
j=1 |zj|<1}.
P o e (o find a coun e example) ha o ∈C
(0,1)(DN)wi h ¯
∂ =0
he e is u∈C(DN)such ha ¯
∂u = and
u∞≤C ∞.
The key poin he e is ha Cshould be independen o N. One can ask
he same ques ion o D2
N={z∈CN;N
j=1 |zj|2<1}[L].
P oblems in Complex Analysis and Dynamics 537
Ques ion 3.4 (Be nd sson).Le φbe a bounded s ic ly plu isubha -
monic unc ion in he uni ball B⊂Cn.Le Ω=i∂ ¯
∂φ, he K¨ahle o m
associa ed o φ.Le be a (0,1) o m on Bsuch ha | |2
Ω+|∂ |Ω≤C.
The subsc ip means ha he no m o and ∂ a e measu ed wi h e-
spec o Ω. Does he equa ion ¯
∂u = ha e a bounded solu ion in B?
The ques ion is ela ed o he Co ona heo em in se e al a iables [Be].
Ques ion 3.5 (Edi ed by Sophia Vassiliadou).Le Xbe a closed ana-
ly ic se o pu e dimension pin Cn+pwi h singula i ies. Le Reg(X)=
X Sing(X)deno e he se o smoo h poin s o X. Gi e Reg(X) he
me ic induced by he imbedding Reg(X)8→Cn+p.
(a) Le ∈L2
(0,q)(Reg(X)),1≤q≤p,¯
∂ =0in he weak sense
in L2
0,q+1(Reg(X)). Find he obs uc ions o sol ing ¯
∂u = in
he weak sense in L2
0,q(Reg(X)). Recall ha ¯
∂u = in he weak
sense in L2
0,q(Reg(X)) i and only i u∈L2
0,q−1(Reg(X)) and o
all ψ∈C
∞
p,p−q(Reg(X)), compac ly suppo ed in Reg(X)we ha e
Reg(X)
u∧¯
∂ψ =(−1)qReg(X)
∧ψ.
(b) Le h∈L2
(α,β)(Reg(X)). We say ha hbelongs o he domain o
¯
∂wi h Di ichle bounda y condi ions —h∈Dom ¯
∂D— i and only
i he e exis hn∈C
∞
α,β(Reg(X)), compac ly suppo ed in Reg(X)
and g∈L2
α,β+1(Reg(X)) such ha hn→hin L2
α,β(Reg(X)) and
¯
∂hn→gin L2
α,β+1(Reg(X)). In ha case we w i e ¯
∂Dh=: g.
Le ∈L2
(p,q)(Reg(X)),1≤q≤psuch ha ¯
∂D =0. Find he
obs uc ions o sol ing ¯
∂Du= , (i.e. obs uc ions o ob aining
un∈C
∞
(p,q−1)(Reg(X)), compac ly suppo ed in Reg(X)such ha
un→uin L2
(p,q−1)(Reg(X)) and ¯
∂un→ in L2
(p,q)(Reg(X))).
In [PS], (a) and (b) a e sol ed when Xis a p ojec i e su ace wi h
isola ed singula i y. They also compu ed obs uc ions o sol ing ¯
∂weakly
o ¯
∂closed (p, q) o ms when Xis a p ojec i e a ie y o dimension p,
as well as obs uc ions o sol ing ¯
∂D o (0,q),¯
∂D-closed o ms again
when Xis a p ojec i e a ie y o dimension p.
See [BeSi] o an al e na i e poin o iew: Sol ing ¯
∂on posi i e
cu en s (especially posi i e, closed cu en s o bideg ee (1,1)).
See [DFV] o obs uc ions o sol ing ¯
∂weakly o (0,1) o ms nea
2-dimensional isola ed singula i ies in Cn,n≥3.
544 J. E. Fo næss, N. Sibony
[FS11] J. E. Fo næss and N. Sibony, Inc easing sequences o com-
plex mani olds, Ma h. Ann. 255(3) (1981), 351–360.
[FS12] J. E. Fo næss and N. Sibony, C i ically fini e a ional maps
on P2, in: “The Madison Symposium on Complex Analysis”
(Madison, WI, 1991), Con emp. Ma h. 137, Ame . Ma h. Soc.,
P o idence, RI, 1992, pp. 245–260.
[FW] J. E. Fo næss and B. Weicke , A ac o s in P2, in: “Se -
e al complex a iables” (Be keley, CA, 1995–1996), Ma h. Sci.
Res. Ins . Publ. 37, Camb idge Uni . P ess, Camb idge, 1999,
pp. 297–307.
[F ] S. F ankel, Complex geome y o con ex domains ha co e
a ie ies, Ac a Ma h. 163(1–2) (1989), 109–149.
[FM] S. F iedland and J. Milno , Dynamical p ope ies o plane
polynomial au omo phisms, E godic Theo y Dynam. Sys ems
9(1) (1989), 67–99.
[Ga] E. A. Ga os o, A ac ing basins in P2,J. Geom. Anal. 8(3)
(1998), 433–440.
[Gh] E. Ghys, Lamina ions pa su aces de Riemann, in: “Dy-
namique e g´eom´e ie complexes” (Lyon, 1997), Pano .
Syn h`eses 8, Soc. Ma h. F ance, Pa is, 1999, pp. 49–95.
[GS] J. Globe nik and B. S ensønes, Holomo phic embeddings
o plana domains in C2,Ma h. Ann. 303(4) (1995), 579–597.
[G ] J. M. G eene, On he las KAM o us o he s anda d map,
J. Ma h. Phys. 20 (1979), 1183.
[G] M. G omo , Pseudoholomo phic cu es in symplec ic mani-
olds, In en . Ma h. 82(2) (1985), 307–347.
[Gu] V. Guedj, App oxima ion o cu en s on complex mani olds,
Ma h. Ann. 313(3) (1999), 437–474.
[Ha1] M. Hakim, Analy ic ans o ma ions o (Cp,0) angen o he
iden i y, Duke Ma h. J. 92(2) (1998), 403–428.
[Ha2] M. Hakim, T ans o ma ions angen o he iden i y, s able
pieces o mani olds, P epublica ions O say.
[HeP] G. M. Henkin and P. L. Polyako , The G o hendieck-
Dolbeaul lemma o comple e in e sec ions, C. R. Acad. Sci.
Pa is Se . I Ma h. 308(13) (1989), 405–409.
[H] S. I. H uska, Ph. D hesis, Co nell, in p epa a ion.
[HO] J. H. Hubba d and R. W. Obe s e-Vo h,H´enon map-
pings in he complex domain. II. P ojec i e and induc i e lim-
i s o polynomials, in: “Real and complex dynamical sys ems”
(Hille ød, 1993), NATO Ad . Sci. Ins . Se . C Ma h. Phys. Sci.
464, Kluwe Acad. Publ., Do d ech , 1995, pp. 89–132.
P oblems in Complex Analysis and Dynamics 545
[HP] J. H. Hubba d and P. Papadopol, Supe a ac i e fixed
poin s in Cn,Indiana Uni . Ma h. J. 43(1) (1994), 321–365.
[HM] S. Hu de and Y. Mi suma su, The in e sec ion p oduc
o ans e se in a ian measu es, Indiana Uni . Ma h. J. 40(4)
(1991), 1169–1183.
[IS] F. M. Iz a˘
ıle and D. L. ˇ
Sepeljanski˘
ı, Quan um esonance
o he o a o in a nonlinea pe iodic field, (Russian), Teo e .
Ma . Fiz. 43(3) (1980), 417–428.
[J] M. Jonsson, Some p ope ies o 2-c i ically fini e holomo phic
maps o P2,E godic Theo y Dynam. Sys ems 18(1) (1998),
171–187.
[JV] M. Jonsson and D. Va olin, P ep in (2001).
[JW] M. Jonsson and B. Weicke , A nonalgeb aic a ac o in
P2,P oc. Ame . Ma h. Soc. 128(10) (2000), 2999–3002.
[JL] D. Jupi e and K. Lilo , P ep in (2001).
[L] L. Lempe , The Dolbeaul complex in infini e dimension. II,
J. Ame . Ma h. Soc. 12(3) (1999), 775–793.
[Ma] N. G. Maka o , Me ic p ope ies o ha monic measu e, in:
“P oceedings o he In e na ional Cong ess o Ma hema icians”,
Vol. 1,2 (Be keley, Cali ., 1986), Ame . Ma h. Soc., P o idence,
RI, 1987, pp. 766–776.
[M] B. Malg ange, Sys `emes diffe en iels `a coefficien s cons an s,
Seminai e Bou baki 15 (1962–63), no. 246.
[Mi] J. Milno ,“Dynamics in one complex a iable: In oduc o y
lec u es”, F ied . Vieweg & Sohn, B aunschweig, 1999.
[N] R. Na asimhan, The Le i p oblem in he heo y o unc ions
o se e al complex a iables, in: “P oc. In e na . Cong . Ma he-
ma icians” (S ockholm (1962), Ins . Mi ag-Leffle , Dju sholm,
1963, pp. 385–388.
[Ni] L. Ni enbe g, A ce ain p oblem o Hans Lewy, (Russian),
Uspehi Ma . Nauk 29(2) (1974), 241–251.
[NWY] J. Noguchi, J. Winkelmann and K. Yamanoi, The alue
dis ibu ion o holomo phic cu es in o semi-abelian a ie ies, C.
R. Acad. Sci. Pa is S´e . I Ma h. 331(3) (2000), 235–240.
[O] S. Yu. O e ko , An example in connec ion wi h he Jacobian
conjec u e, (Russian), Ma . Zame ki 47(1) (1990), 127–136, 173;
ansla ion in Ma h. No es 47(1–2) (1990), 82–88.
[PS] W. L. Pa don and M. A. S e n,L2-¯
∂-cohomology o com-
plex p ojec i e a ie ies, J. Ame . Ma h. Soc. 4(3) (1991),
603–621.
[PM] R. P´
e ez-Ma co, Siegel discs wi h smoo h bounda y, P ep in .
546 J. E. Fo næss, N. Sibony
[Pi] S. I. Pinchuk, On he analy ic con inua ion o holomo phic
mappings, Ma . Sb. 27 (1975), 375–392.
[R] M. Rees, Posi i e measu e se s o e godic a ional maps, Ann.
Sci. ´
Ecole No m. Sup. (4) 19(3) (1986), 383–407.
[Ru] D. Ruelle,“Elemen s o diffe en iable dynamics and bi u ca-
ion heo y”, Academic P ess, Inc., Bos on, MA, 1989.
[Sa] F. Sa kis, Ha ogs-Bochne ype heo em in p ojec i e space,
P ep in (2001).
[Sh] R. Sha iko , Analy ic con inua ion o ge ms o holomo phic
mappings be ween eal hype su aces in Cn,Michigan Ma h. J.
47(1) (2000), 133–149.
[S1] N. Sibony, Dynamique des applica ions a ionelles de Pk,
in: “Dynamique e g´eom´e ie complexes” (Lyon, 1997), Pano .
Syn h`eses 8, Soc. Ma h. F ance, Pa is, 1999, pp. 97–185.
[S2] N. Sibony, Une classe de domaines pseudocon exes, Duke Ma h.
J. 55(2) (1987), 299–319.
[Si1] Y.-T. Siu, Pseudocon exi y and he p oblem o Le i, Bull.
Ame . Ma h. Soc. 84(4) (1978), 481–512.
[Si2] Y.-T. Siu,¯
∂ egula i y o weakly pseudocon ex domains in
compac He mi ian symme ic spaces wi h espec o in a ian
me ics, P ep in (2001).
[SY] Y.-T. Siu and S.-K. Yeung, Hype bolici y o he complemen
o a gene ic smoo h cu e o high deg ee in he complex p ojec-
i e plane, In en . Ma h. 124(1–3) (1996), 573–618.
[S ] B. S ensønes, Fa ou-Biebe bach domains wi h C∞-smoo h
bounda y, Ann. o Ma h. (2) 145(2) (1997), 365–377.
[T] M. Tsuji,“Po en ial heo y in mode n unc ion heo y”,
Ma uzen Co., L d., Tokyo, 1959.
[U] T. Ueda, C i ical o bi s o holomo phic maps on p ojec i e
spaces, J. Geom. Anal. 8(2) (1998), 319–334.
[Vi] C. Vi e bo, Func o s and compu a ions in Floe homology
wi h applca ions. I, Geom. Func . Anal. 9(5) (1999), 985–1033.
[V] A. G. Vi ushkin, Some examples ela ed o he p oblem o
polynomial ans o ma ions o Cn,Ma h. USSR-Iz . 5(1971),
269–279.
[We] B. Weicke , Infini e-dimensional complex dynamics: a quan-
um andom walk, Disc e e Con in. Dynam. Sys ems 7(3)
(2001), 517–524.
[W] T. H. Wol , Plane ha monic measu es li e on se s o σ-fini e
leng h, A k. Ma . 31(1) (1993), 137–172.
P oblems in Complex Analysis and Dynamics 547
John E ik Fo næss:
Depa men o Ma hema ics
The Uni e si y o Michigan
Eas Hall, Ann A bo , Mi 48109
U.S.A.
E-mail add ess:[email p o ec ed]
Nessim Sibony:
CNRS UMR8628
Depa men o Ma hema ics
Uni e si ´e Pa is-Sud
Ba imen 425
O say Cedex
F ance
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 26 de eb e de 2001,
da e a e si´o ebuda el 14 de ma ¸c de 2001.
