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Uni e sal en i e unc ions o a ine
endomo phisms o CN
L. Be nal-Gonz´alez
Abs ac
In his pape he a ine endomo phisms o CNwhich suppo compo-
si ionally uni e sal en i e unc ions a e comple ely cha ac e ized.
1 In oduc ion
Th oughou his pape Nwill deno e he se o all posi i e in ege s, and i
N∈N hen CNwill s and o he N-dimensional complex space. In pa icu-
la , C1is he complex plane C. The closed polydisk o adius ≥0 cen e ed
a he o igin is D( ) = {z= (z1, . . . , zN)∈CN:kzk ≤ }, whe e kzk=
max1≤j≤N|zj|. A domain Go CNis a nonemp y connec ed open subse o
CN. By H(G) we deno e he space o holomo phic unc ions :G→C,
endowed wi h he opology o uni o m con e gence in compac a. In pa i-
cula , H(CN) is he space o en i e unc ions o Ncomplex a iables. I is
well known ha H(G) becomes a sepa able F ´eche space unde he abo e
opology. The symbol Au (G) will s and o he g oup o all au omo phisms
(= biholomo phic bijec i e sel mappings) on G.
In 1929 Bi kho [8] cons uc ed an en i e unc ion which is ‘uni e sal’ o
ansla ions. In ac , he p o ed essen ially ha gi en b∈C {0} he e exis s
∗Pa ially suppo ed by Plan Andaluz de In es igaci´on Jun a de Andaluc´ıa FQM-127.
2000 Ma hema ics Subjec Classi ica ion: P ima y 47A16. Seconda y 32E30, 47B38.
Key wo ds and ph ases: uni e sal en i e unc ion, composi ion ope a o , endomo phism
o CN, holomo phic con exi y.
Au ho ’s add ess: Depa amen o de An´alisis Ma em´a ico. A da. Reina Me cedes,
Apdo. 1160. 41080-Se illa, Spain. E-mail: lb[email p o ec ed].
1
a unc ion ∈H(C) such ha i s sequence o ansla es { (·+nb) : n∈N}
is dense in H(C).
Bi kho ’s heo em can be obse ed unde he poin o iew o he ope a o
heo y as a uni e sali y esul ; namely, i ϕ:C→Cdeno es he ansla ion
z7→ z+b hen he composi ion ope a o
Cϕ: ∈H(C)7→ ◦ϕ∈H(C)
is uni e sal. In gene al, i Xis a (necessa ily sepa able) opological ec o
space and Tis an ope a o (= con inuous linea sel mapping) on X hen
(Tn) is said o be uni e sal (o hype cyclic) p o ided ha he e exis s some
ec o x∈X–called uni e sal o T– o which he o bi {Tnx:n∈N}
o xunde Tis dense in X. He e (Tn) ep esen s he sequence o i e a es
T1=T, T2=T◦T, . . . o T. I is easy o see ha he se o uni e sal
ec o s is dense. The ope a o Tis called he edi a ily uni e sal i and only
i gi en a sequence {n1< n2<· · · } ⊂ N he e is a ec o x∈Xsuch
ha {Tnkx:n∈N}is dense in X. I Xis Bai e and me izable and Tis
uni e sal (he edi a ily uni e sal) hen he se o uni e sal ec o s o T( o
each sequence (Tnk), espec i ely) is esidual, ha is, i s complemen is o
i s ca ego y. These no ions can be easily ex ended o a sequence (Tn) o
ope a o s. See [16] o a good accoun abou hese concep s and hei his o y.
Since 1929 many pape s ha e deal wi h he subjec o uni e sali y h ough
ansla ions in one (complex) a iable, bu only a ew ones in se e al a i-
ables. Le us make a b ie epo , now in he language o he uni e sali y o
ope a o s; see also he su ey [16] –specially i s Sec ion 4a– which con ains
a a he comple e lis o e e ences including domains G6=Cand spaces
X6=H(G). In 1976 Luh [21] p o ed ha o a p esc ibed unbounded se-
quence (bn)⊂C he sequence (Cϕn) is uni e sal on H(C), whe e ϕnis he
ansla ion z7→ z+bn. In 1984 Duyos-Ruis [11] showed by unc ional analy-
sis me hods ha Cϕ(ϕ(z) = z+b, b ∈C {0}) is uni e sal on H(C) (hence
he e is a esidual subse o uni e sal unc ions), while he esiduali y o he
(Cϕn)-uni e sal en i e unc ions (whe e he ϕna e he abo e ansla ions)
was obse ed by G osse-E dmann [15] and Ge hne and Shapi o [12]. In
1995 Be nal and Mon es [6] we e able o show he same esul o a sequence
{ϕn(z) = anz+bn:n∈N} ⊂ Au (C); ecall ha ϕ∈Au (C) i and only i
ϕis a noncons an a ine endomo phism o C, ha is, he e a e a, b ∈Cwi h
a6= 0 and ϕ(z) = az +b. Speci ically, hey p o ed ha (Cϕn) is uni e sal
2
i and only i he sequence {min{|bn|,|bn/an|} :n∈N}is unbounded, om
which hey de i ed ha i ϕis an a ine endomo phism o C hen
Cϕis uni e sal i and only i ϕis a ansla ion,
ha is, ϕ(z)≡z+b o some b∈C {0}. As o se e al a iables, he his o y
is no oo long. In 1941 Seidel and Walsh [23] p o ed a non-Euclidean e sion
o Bi kho ’s heo em o he uni disk, and in 1979 Chee [10] ex ended his
o he uni polidisk and ball o CN. Le´on [20] has ecen ly cha ac e ized he
co esponding uni e sal sequences o au omo phisms in bo h domains. In
he case o he euclidean ansla ions ϕ(z) = z+b(z∈CN, b ∈C {0})
in se e al a iables –whe e z= (z1, . . . , zN), b = (b1, . . . , bN), and hese will
be he s anda d ep esen a ions o z, b along he cu en pape – he na u al
ex ensions o he heo ems o Bi kho and Luh a e co e ed by [13, Sec ion
5] and by he ecen pape s [1], [3, Sec ion 2] and [7] (see also [2] and [3] o
a ma icial ex ension o Zappa’s esul [24], which in u n is a mul iplica i e
e sion in C {0}o Bi kho ’s heo em).
In iew o he abo e disco e ies, i is na u al o pose he p oblem o
cha ac e izing hose mappings ϕ∈Au (CN) such ha Cϕis uni e sal on
H(CN). Ne e heless, a comple e desc ip ion o Au (CN) (see o ins ance
[4], [5] and [22] o a s udy o some sub amilies o i ) is unknown up o da e.
Al hough he e a e plen y o au omo phisms o CN, he simples among hem
a e wi h no doub he a ine linea mappings (o ‘a ine endomo phisms’) om
CNin o i sel which a e in e ible. Each a ine endomo phism S=S(A, b) is
biuni ocally de e mined by a pai (A, b), whe e A:= [aij]i,j=1,...,N is a ma ix
wi h complex en ies and bis a ixed ec o o CN; so Sis gi en by
S(z) = Az +b o all z∈CN.
Obse e ha as we make he calcula ion S(z) = Az +bi is con enien o
conside he ec o s zand bas ‘column’ ec o s. I is clea ha S∈Au (C)
i and only i Sis one- o-one i and only i Sis on o i and only i de (A)6= 0.
Hence he main aim o his pape is o cha ac e ize he uni e sali y o
he composi ion ope a o CS:H(CN)→H(CN) gene a ed by an a ine
endomo phism S=S(A, b) in e ms o he ma ix Aand he ec o b. This
will be accomplished in Sec ion 3 whe e we p o e, among o he hings, ha
CSis uni e sal i and only i Sis uni alen and has no ixed poin . In Sec ion
2 we p esen a numbe o s a emen s ha will e eal use ul o ou goal.
3
2 Se e al auxilia y esul s
This sec ion is de o ed o backg ound ma e ial on N-dimensional complex
app oxima ion ha will be needed o he wo k o Sec ion 3.
F om now on Gwill ep esen a domain in CN. Le us deno e by H(G, G)
he se o all holomo phic sel mappings ϕ= (ϕ1, . . . , ϕn) on G, ha is,
ϕ(G)⊂Gand each componen ϕj:G→C(j= 1, . . . , N) is a holomo phic
unc ion. Then i ϕ∈H(G, G) he composi ion ope a o Cϕ:H(G)→H(G)
is well de ined. O cou se, Au (G)⊂H(G, G). Ou i s lemma p e en s us
o use non-injec i e sel mappings o ob ain Cϕ-uni e sali y.
Lemma 2.1. I ϕ∈H(G, G)and Cϕis uni e sal on H(G) hen ϕis one-
o-one and has no ixed poin s.
P oo . The esul s con ained in his lemma a e well known in he one-dimen-
sional con ex , see o ins ance [9, pages 3 and 10]. The p oo gi en on page
10 o ha monog aph o he necessi y o uni alence wo ks, wo d o wo d, in
se e al a iables. Indeed, i ϕiden i ies wo dis inc poin s aand bo G, hen
so does he n- h componen ϕno ϕ, and so does ◦ϕn o each nand each
∈H(G). Thus i gis a limi poin o he Cϕ-o bi o , hen g(a) = g(b),
hence (because some g∈H(G), namely an app op ia e coo dina e unc ion,
akes di e en alues a aand b) no holomo phic on Gcan include e e y
unc ion in H(G) in he closu e o i s o bi . Hence Cϕis no uni e sal.
Finally, i a∈Gwe e a ixed poin o ϕand ∈H(G) we e Cϕ-uni e sal
hen by conside ing he compac se K={a} he closu e in CNo he se
{ (ϕn(a)) : n∈N}={ (a)}would be dense in C, which is absu d.
A se B⊂CNis said o be H(CN)-con ex (see [17] o [19]) whene e
e
B=B, whe e
e
B:= {z∈CN:| (z)| ≤ sup
∈B
| ( )| o all ∈H(CN)}.
The nex gene aliza ion o Runge’s app oxima ion heo em o se e al com-
plex a iables is a special case o a s a emen ha can be ound in [17,
Theo em 4.3.2 and ollowing no e].
P oposi ion 2.2. Le be a holomo phic unc ion in a neighbo hood o an
H(CN)-con ex compac subse Ko CN. Then he e is a sequence ( j)⊂
H(CN)such ha j→ (j→ ∞)uni o mly on K.
4

In 1965 Kallin [18] p o ed an impo an sepa a ion lemma in se e al a i-
ables, om which he ollowing p oposi ion – ha will be c ucial o ou ap-
p oxima ion p oblem– is a pa icula case. The wo d “con ex” means “ge-
ome ically con ex”, ha is, a se B⊂CNis con ex whene e z, w ∈B
implies λz + (1 −λ)w∈B o all λ∈[0,1].
P oposi ion 2.3. I Kand La e disjoin con ex compac se s in CN hen
K∪Lis H(CN)-con ex.
In connec ion wi h he las p oposi ion we poin ou ha i is no known
ye whe he he disjoin union o 4 closed balls in CNis H(CN)-con ex.
The ollowing lemma se les he ques ion o which sequences o au omo -
phisms a e adequa e o gene a e uni e sali y. Following [6], we say ha a
sequence (ϕn)⊂H(G, G) is un-away i and only i gi en a compac se
K⊂G he e exis s n0=n0(K)∈Nsuch ha K∩ϕn0(K) = ∅; and we say
ha a unc ion ϕ∈H(G, G) is non- ecu en whene e i s sequence (ϕn) is
un-away.
Lemma 2.4. Suppose ha ϕis an a ine au omo phism o CN. Then Cϕis
uni e sal on H(CN)i and only i ϕis non- ecu en .
P oo . Le us suppose ha Cϕis uni e sal and ha , by way o con adic ion,
ϕis no non- ecu en . Then he e is a compac se Ksuch ha K∩ϕn(K)6=
∅ o all n∈N. Choose a sequence (zn)⊂Kwi h (ϕn(zn)) ⊂Kand a
Cϕ-uni e sal unc ion ∈H(G). Conside he cons an unc ion g(z) =
1 + maxK| |. We ha e ha , o e e y n∈N,
max
z∈K|g(z)− (ϕn(z))| ≥ ||g(z)|−| (ϕn(zn))|| = 1+max
K| |−| (ϕn(zn))| ≥ 1,
which is a con adic ion.
Con e sely, assume ha ϕis non- ecu en . Ou inal goal is o show
ha he se Mo uni e sal unc ions o Cϕis esidual (so nonemp y). Since
H(G) is a second-coun able Bai e space, Bi kho ’s ansi i i y heo em (see
o ins ance [14, 9.20]) asse s ha Mis a dense Gδ-subse (hence esidual)
i and only i o e e y pai o nonemp y open subse s Aand Bo H(CN)
he e exis s some m∈Nwi h
(Cϕ)m(A)∩B=∅.(1)
5
No e ha (Cϕ)n=Cϕn(n∈N). Fix A, B as be o e. Then he e exis s
ε > 0, R > 0 and , h ∈H(CN) such ha A⊃A1:= {g∈H(CN) :
maxz∈D(R)|g(z)− (z)|< ε}and B⊃B1:= {g∈H(CN) : maxz∈D(R)|g(z)−
h(z)|< ε}. Since ϕis non- ecu en , he e exis s m∈Nsuch ha D(R)∩
ϕm(D(R)) = ∅. Bu ϕm(D(R)) is a con ex compac se because ϕmis
con inuous and con ex-p ese ing ( ha is, ϕm(C) is con ex whene e Cis
con ex; his is ue because ϕis a ine). Then L:= D(R)∪ϕm(D(R)) is
H(CN)-con ex by P oposi ion 2.3. Fix Uand Vopen subse s in CNsuch
ha D(R)⊂U,ϕm(D(R)) ⊂V(so U∪V⊃L) and U∩V=∅. De ine he
unc ion F:U∩V→Cas
F(z) =  (z) i z∈U
h(ϕ−m(z)) i z∈V,
which is holomo phic on U∪V. Hence by P oposi ion 2.2 he e exis s an
en i e unc ion gsa is ying
|F(z)−g(z)|< ε o all z∈L.
Bu om he de ini ion o Fwe ge
|g(z)− (z)|< ε o all z∈D(R).
and
|g(z)−h(ϕ−m(z))|< ε o all z∈ϕm(D(R)).
The las display is clea ly equi alen o
|g(ϕm(z)) −h(z)|< ε o all z∈D(R).
In o he wo ds, g∈A1and (Cϕ)mg∈B1. Thus, g∈Aand (Cϕ)mg∈B, so
(1) holds.
Rema k 2.5. The p oo o Lemma 2.4 can be easily modi ied o ob ain he
ollowing ex ension: Suppose ha Gis a con ex domain o CNand ha
(ϕn) is a sequence in Au (G) o con ex-p ese ing mappings. Then (Cϕn) is
uni e sal i and only i (ϕn) is un-away. Mo eo e , in his case he e exis s
a esidual subse o uni e sal unc ions. Su ice o say ha i Gis con ex
hen he polidisks D(R) o he p oo o Lemma 2.4 can be eplaced o con ex
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compac subse s o Gand ha Bi kho ’s ansi i i y heo em also wo ks
wi h a sequence (Tn) o mappings om a Bai e space in o a second-coun able
space, see [16, Theo em 1]. I migh be in e es ing o in es iga e whe he
he las lemma can be ex ended o non-con ex domains o o sequences o
au omo phisms ha do no p ese e con exi y.
3 Uni e sal unc ions o endomo phisms
F om now on Awill ep esen an (N×N)-ma ix wi h complex en ies
and bwill be a ixed ec o in CN.
Lemma 2.4 ocuses a en ion on he dynamics o a ine mappings o CN.
In o de o cha ac e ize when hey gene a e uni e sal composi ion ope a o s,
we need o know which o such mappings a e non- ecu en . We a e now
eady o s a e ou main esul .
Theo em 3.1. Assume ha S:CN→CNis an a ine endomo phism, say
Sz =Az +b(z∈CN). Conside he composi ion ope a o CSgene a ed by
S. Then he ollowing p ope ies a e equi alen :
(a) Shas no ixed poin in CNand de (A)6= 0.
(b) The ec o bis no in an(A−I)and de (A)6= 0.
(c) CSis uni e sal.
(d) CSis he edi a ily uni e sal.
P oo . (a) ⇐⇒ (b): Simply obse e ha b∈ an(A−I) i and only i he e
exis s z0∈CNsuch ha (A−I)z0=bi and only i Az0+b=z0 o some
z0∈CNi and only i Sz0=z0 o some z0∈CN.
(d) =⇒(c): This is i ial.
(c) =⇒(a): I CSis uni e sal hen Sis one- o-one (hence de (A)6= 0) and
has no ixed poin by Lemma 2.1.
(b) =⇒(d): Since de (A)6= 0, Sis an a ine au omo phism o CN. Le us
p o e ha Sis non- ecu en . Deno e by J he Jo dan ma ix o A. Then
he e is an in e ible (N×N)-ma ix Qsuch ha A=QJQ−1. De ine
c:= Q−1band Mz := Jz +c(z∈CN).
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I is easy o see ha b6∈ an(A−I) i and only i c6∈ an(J−I). On
he o he hand, non- ecu ence is p ese ed by simila i ies; mo e p ecisely,
i ϕ∈H(CN,CN) and ψ∈Au (CN), hen ϕis non- ecu en i and only
ψ◦ϕ◦ψ−1is non- ecu en . Since S=Q◦M◦Q−1we ob ain ha Sis
non- ecu en i and only i Mis.
The ma ix Jis a di ec sum o Jo dan blocks Jj. The ec o chas
componen s co esponding o each o hese blocks and o say c6∈ an(J−I)
is o say ha a leas one o hese componen , say cjis no in an(Jj−Ij).
He e Ijis he iden i y ma ix ha ing he same dimension as Jj. Le Mjbe
he es ic ion o M o he di ec summand o CNon which Jjac s, i.e.,
Mjzj=Jjzj+cj. I su ices o p o e ha Sjis non- ecu en ( his ollows
om he ac ha i CNis ep esen ed as a p oduc space, hen each compac
subse o CNis con ained in a p oduc o compac subse s co esponding o
he ac o s o CN). Hence we may d op he subsc ip s and assume ha J
i sel is a Jo dan block wi h c6∈ an(J−I). In pa icual , J−Iis no
in e ible. Thus he spec um o Jis he single on {1}, so Ji sel has jus
1’s on he main diagonal and he i s supe diagonal, and ze os elsewhe e.
Bu , by induc ion, one ob ains
Mnz=Jnz+
n−1
X
k=0
Jkc(z∈CN, n ∈N).
The “1” in he (N, N) posi ion is c ucial he e; i is also he (N, N) en y o any
powe o M, and in all hese powe s he es o he N- h ow consis s o ze os.
Thus (Mnz)N, he N- h componen o Mnz, is zN+ncN. Now c6∈ an(J−I)
means cN6= 0, hence as n→ ∞ he N- h componen o Mnzgoes o in ini y
uni o mly on each compac subse o CN. Hence ||Snz|| → +∞(n→ ∞) in
he same way. I is easy o see ha because o his Mis ecu en .
Consequen ly, Sis non- ecu en and, due o Lemma 2.4, CSis uni e sal.
Finally, obse e ha i we ix a sequence {n1< n2<· · · } ⊂ N hen he
same easoning abo e – eplacing n o nj– shows ha (Mnj) (hence (Snj))
is un-away, which by Rema k 2.5 p o es ha (CSnj) is uni e sal. In o he
wo ds, CSis he edi a ily uni e sal.
Rema ks 3.2. 1. The p oo o Theo em 3.1 e eals ha , e en in he case
ha Sis no in e ible, we ha e: Shas no ixed poin i and only i b6∈
8