Publ. Ma . 48 (2004), 227–239
ON COMMUTING POLYNOMIAL AUTOMORPHISMS
OF C2
Cinzia Bisi∗
Abs ac
We cha ac e ize he commu ing polynomial au omo phisms o C2,
using hei me omo phic ex ension o P2and looking a hei dy-
namics on he line a in ini y.
1. In oduc ion
The g oup o polynomial au omo phisms o C2, Au (C2), consis s o
bijec i e maps:
: (z, w)∈C2→( 1(z, w), 2(z, w)) ∈C2
whe e 1, 2∈C[z, w].
When is polynomial and bijec i e, hen he in e se −1is polyno-
mial.
Following [4], we in oduce wo subg oups o Au (C2), he g oup Eo
elemen a y maps
E={(z, w)→(αz +p(w), βw +γ) : α, β, γ ∈C, αβ 6= 0, p ∈C[w]}
and he g oup Ao a ine maps
A={(z, w)→(a1z+b1w+c1, a2z+b2w+c2) : ai, bi, ci∈C, a1b2−a2b16=0}.
An elemen a y map p ese e he ho izon al olia ion dw = 0.
We deno e by AT =A ∩ E he g oup o he au omo phisms a ine
and iangula , i.e.:
AT ={(z, w)→(a1z+b1w+c1, b2w+c2) : a1, bi, ci∈C, a1b26= 0}.
2000 Ma hema ics Subjec Classi ica ion. P ima y: 32H50, 14R10; Seconda y:
37F10, 58F23.
Key wo ds. H´enon maps, inde e minacy poin s, G een unc ions, illed Julia se .
∗Pa ially suppo ed by P oge o MURST di Rile an e In e esse Nazionale P op ie `a
geome iche delle a ie `a eali e complesse.
228 C. Bisi
We ecall now a heo em on he s uc u e o Au (C2) which is known
only in dimension 2. I is due o Jung, [5]; i was ep o ed in se e al
di e en ways [9] and ecen ly also in [8]. Jung’s Theo em asse s ha
he g oup Au (C2) is he amalgama ed p oduc o i s subg oups Eand
Awi h espec o hei in e sec ion AT . By his heo em, each au-
omo phism ϕ∈Au (C2)− AT can be w i en as a composi ion o
elemen a y and a ine au omo phisms which can s a o inish indi e -
en ly wi h an a ine o an elemen a y map.
A ini e composi ion o maps o he o m:
hj(z, w) = (pj(z)−ajw, z) = (−ajz+pj(w), w)◦(w, z) = ej◦a
(whe e aj∈C∗,pjis a polynomial o deg ee dj≥2, ej∈ E,a∈ A and
i is he in e sion o he coo dina es) is called a H´enon map.
The se o H´enon maps is a semig oup and i is deno ed by H.
P oposi ion 1.1. [4]A polynomial au omo phism o C2is conjuga e,
in he g oup o polynomial au omo phisms, o an elemen a y map o o
a map in H.
Le = ( 1, 2) be a polynomial au omo phism o C2o algeb aic
deg ee d≥2. We will deno e by i s me omo phic ex ension o P2.
The g aph Γ o is he closu e in P2o he g aph o . Le (z, w) be
a ine coo dina es in C2and le [z:w: ] be co esponding homogeneous
coo dina es in P2, hen he line a in ini y L∞has equa ion { = 0}.
We will deno e espec i ely I+and I− he inde e minacy subse s o
and o −1. These a e wo analy ic subse s o codimension a leas 2
in P2, con ained in L∞. I is known, [11, p. 106], ha hey a e bo h
composed by a mos one poin . I pis an inde e minacy poin , we de ine
(p) as he analy ic subse o Γ which p ojec s on p, i coincides wi h
∩>0 (B(p, )−I); we call (p) he blow-up a p.
De ini ion 1.2. [11] A polynomial au omo phism is egula i I+( )6=
I−( ).
The H´enon maps a e egula , whe eas o elemen a y maps we ha e
I+=I−.
Obse e ha he no ion depends on choice o coo dina es.
We s udy, in his pape , he equa ion ◦g=g◦ o polynomial
au omo phisms o C2. The i s esul ha we will p o e is he ollowing
Main Lemma:
On Comm. Polyn. Au om. o C2229
Lemma 1.3. Suppose ha ,ga e wo commu ing polynomial au omo -
phisms o C2, no o a ine ype, hen a leas one o he wo ollowing
cases occu s:
(i) I+
=I+
g(which implies also I−
=I−
g);
(ii) I+
=I−
g(which implies also I−
=I+
g).
As a consequence o i , we ha e ha a egula map canno commu e
wi h a non a ine elemen a y map. We ge :
P oposi ion 1.4. Le CA( )be he g oup o a ine au omo phisms o C2
which commu e wi h . I is egula , hen CA( )is a ini e cyclic
subg oup o A.
Theo em 1.5. Le ,gbe wo egula au omo phisms o C2 espec i ely
o deg ee d1and d2. Suppose ha ◦g=g◦ . Then he e exis n0, m0∈
Nsuch ha dn0
1=dm0
2and he e exis s an a ine au omo phism hsuch
ha n0=gm0◦h.
P oposi ion 1.4 and Theo em 1.5 we e p o ed by Lamy, [6], [7] using
he ac ion o Au (C2) on he ee whose e ices a e he cose s o he
subg oups Aand E; un o una ely his ac ion can be de ined only when
he g oup is an amalgama ed p oduc , [10], hence Lamy’s app oach de-
pends on Jung’s s uc u e heo em and i canno be gene alized o highe
dimensions. Since he analogue o Jung’s Theo em is no a ailable in
highe dimension, we ha e in oduced a new app oach. We hink ha
he app oach we ollow he e will gi e he cen alize o a egula poly-
nomial au omo phism in highe dimension.
Abou commu ing elemen a y maps, i s W igh , [14], p o ed ha
he g oup gene a ed by wo commu ing elemen a y maps con ains Z⊕Z,
hen Lamy, [7], men ioned ha his g oup is no coun able.
Acknowledgemen s. The au ho is e y g a e ul o N. Sibony and
C. Fa e: indeed he i s one has sugges ed he in e es o a new ap-
p oach o he p oblem, he second one has sugges ed a new app oach o
P oposi ion 1.4 and Theo em 1.5.
2. Cha ac e iza ion o commu ing polynomial
au omo phisms o C2
We s a ecalling he ollowing p elimina y esul :
230 C. Bisi
P oposi ion 2.1. [11]I is a non-a ine polynomial au omo phism
o C2, hen
(L∞−I+) = I−;
−1(L∞−I−) = I+.
This is an immedia e consequence o he ollowing elemen a y p op-
e y:
Suppose ha Fand F−1a e he li s o and −1 o C3, hen
F◦F−1(x, y, ) = F−1◦F(x, y, ) = d2−1(x, y, )
whe e d= deg( ) = deg( −1).
P oo o Lemma 1.3: We show i s ha i I+
=I+
g hen I−
=I−
g.
Suppose by con addic ion, ha I−
6=I−
g. Then
(i) ei he I−
6=I+
g;
(ii) o I−
g6=I+
.
In case (i), I−
6=I+
g=I+
hence is egula . In case (ii), I−
g6=I+
=I+
g
hence gis egula . Hence up o change and g, we can suppose ha
I−
6=I+
gand ha is egula . We know ha he closu e o he se K−
in e sec s he line a in ini y only in one poin I−
which is di e en
om I−
g. The e o e i exis s a leas one poin z∈C2such ha z∈K−
bu g(z)/∈K−
. Hence he sequence { −n(z)}is bounded, and also he
sequence g◦ −n(z) is bounded; on he con a y he sequence −n◦g(z)
is no bounded: his con adic s ha −n◦g=g◦ −n. Assume now
ha I+
6=I+
g. Then we ha e:
∀q∈L∞− {I+
, I+
g, I−
, I−
g},
−1◦g−1(q) = −1(I+
g) = I+
(2.1)
unless
I+
g=I−
,(2.2)
and
g−1◦ −1(q) = g−1(I+
) = I+
g
(2.3)
unless
I+
=I−
g.(2.4)
On Comm. Polyn. Au om. o C2231
By he commu a ion p ope y o wi h g, we ha e:
−1◦g−1(q) = g−1◦ −1(q)
hence, excep o he wo cases (2.2) and (2.4), we ha e I+
=I+
g, which
is in con adic ion wi h he assump ion. The e o e we ha e I+
g=I−
o
I+
=I−
g. Bu i u ns ou ha one o he ela ions implies he o he
one by an a gumen simila o he s a ing one.
Lemma 1.3 allows us o assume in he es o he pape ha we a e
in case (i).
Co olla y 2.2. A non a ine elemen a y map canno commu e wi h a
egula one.
The p oo ollows immedia ely om Lemma 1.3.
In Co olla y 2.2 he sys em o coo dina es is ixed, indeed one can
gi e an example o a egula map ha a e conjuga ion i is no mo e
egula , [11].
Example 2.3. I (z, w) = (z2+aw, z) wi h a6={0}, hen is egula .
Le h(z, w) = (w, z +w2), hen g=h−1◦ ◦his no mo e egula .
Co olla y 2.4. Suppose ha and ga e wo commu ing polynomial
au omo phisms o C2, whe e is no o a ine ype. I is conjuga e o
a egula map, hen he same holds o g, in he same coo dina es, o g
is a ine.
P oo : By hypo hesis, he e exis s an au omo phism ρsuch ha ˜
=
ρ◦ ◦ρ−1is egula . Since ˜g=ρ◦g◦ρ−1commu es wi h ˜
hen ˜gis
egula o a ine.
P oo o P oposi ion 1.4: Recall, [11, p. 132], ha a egula biholomo -
phism has in ini ely many dis inc pe iodic o bi s ( his ollows om
Bezou Theo em), no sub a ie y o dimension g ea e o equal han 1 is
pe iodic.
Fi s we wan o p o e ha all he pe iodic poin s o canno lie
on he same complex line. Suppose on he con a y ha he e exis s
a complex line Lsuch ha Sn∈ZFix( n)⊂L(indeed a pe iodic poin
o is a pe iodic poin also o −1o he same pe iod). O cou se
L6=L∞and Lis a he same ime -in a ian and −1-in a ian . Le
{p}=L∩L∞, hen phas o be equal o I−, because (I−) = I−, and i
has also o be equal o I+because −1(I+) = I+. Since I+6=I−, his
is a con adic ion.
232 C. Bisi
I his a ine and ◦h=h◦ , hen, o all N∈N,hinduces a
pe mu a ion on Fix( N) = {pe iodic poin s o o de N o }. So we
ha e a g oup homeomo phism ϕ om CA( ) in o he g oup ΣNo he
pe mu a ions o he poin s o Fix( N).
ϕ:CA( )→ΣN.
I Nis la ge enough, he poin s o Fix( N) do no lie on he same line
and hence ϕis injec i e (an a ine map canno ix mo e han 5 poin s no
on he same line). Hence CA( ) is a ini e g oup o a sui able o de p.
To p o e he cyclici y o CA( ), we p o e ha :
(1) CA( ) is abelian.
(2) The eigen alues o he linea pa o each a ine au omo phism h∈
CA( ) a e oo s o uni y o he same o de .
(3) Fo all h1, h2∈CA( ) o he same o de q, he e exis n0, m0∈N
such ha hn0
1=h2and hm0
2=h1.
In o de o p o e (1), we ecall ha i h◦ = ◦h, hen h(I−
) = I−
and h(I+
) = I+
. Then,up o conjuga ion, we can assume ha I−
= [1 :
0 : 0] and I+
= [0 : 1 : 0].
In hese coo dina es
(2.5) h([x:y: ]) = [αx +γ :βy +δ : ].
Conside now he commu a o [h1, h2] o wo maps h1, h2∈CA( ), hen
i s linea pa in C2is he iden i y 2×2 ma ix, because he linea pa o
each o hem is diagonal, see (2.5). Bu [h1, h2] canno be a ansla ion
o C2because CA( ) is a ini e g oup. Hence he only possibili y is
[h1, h2] = Id.
Since CA( ) is abelian, i ollows ha all he elemen s in CA( ) ha e a
common ixed poin , hence, up o conjuga ion, we can suppose ha hey
a e all o a ions ixing he o igin, he e o e hey a e o ype (αz, βw).
In o de o p o e (2), we ecall ha , since he o de o he g oup CA( )
is p, hen o all h∈CA( ) he e exis s k∈Nwhich di ides psuch ha
hk= Id. This means ha αk=βk= 1, and he eigen alues o h
a e k- oo s o uni y. Bu suppose ha hey ha e di e en o de s, hen
he e exis s a n∈Nwhich di ides ksuch ha hnis he iden i y in
one componen bu no in he o he one. Suppose ha αn= 1 and
βn6= 1. This means ha all he poin s (z, 0) a e ixed by hn. Since o
all m∈Z, mcommu es wi h hn, he line {w= 0}is in a ian o all m,
wi h m∈Z. Fo he in a iance o he line {w= 0}by and by −1, i
ollows ha he unique poin p={w= 0} ∩ L∞has o be equal o I+
and a he same ime o I−
, bu his con adic s he egula i y o .
On Comm. Polyn. Au om. o C2233
The asse ion in (3) ollows di ec ly om (1) and (2): since he o de q
o he o a ion is exac ly he common o de o i s eigen alues, he e exis
an0∈Nsuch ha hn0
1◦h−1
2has an eigen alue equal o 1. Bu hn0
1◦h−1
2
is s ill an elemen in CA( ) and hence i s wo eigen alues ha e he same
o de ; his implies also ha he second eigen alue has o be equal o 1
and hn0
1=h2.
The cyclici y o he g oup CA( ) ollows om (1), (2), (3). I h0is one
o he elemen s o CA( ) o maximal o de s≤p, hen hh0i=CA( ).
Indeed o each h∈CA( ), he o de o hhas o be a di iso o he
maximal o de s; hence he e exis s an elemen in hh0i,h
0, which has
he same o de o h, bu , by (3), his a powe o h
0and so h∈ hh0i. In
conclusion CA( ) is isomo phic o Zp.
We ecall wo examples, see [7], o show ha i is possible o cons uc
ei he egula maps such ha some elemen in CA( ) has wo equal
eigen alues, o egula maps such ha some elemen in CA( ) has wo
di e en eigen alues, bu o he same o de .
Example 2.5. 1) Conside = (y, yn+1 +x). Le αbe equal o β
and αn= 1, hen h= (αx, βy) commu es wi h .
2) Conside = (y, yp+x) and g= (y, yq+x). Le αbe di e en
om βbu αp=βand βq=α, hen h= (αx, βy) commu es
wi h ◦g.
We now p o e Theo em 1.5. We ecall ha , [11], i is a egula
polynomial au omo phism o C2, we can associa e o i he se s:
K+={z∈C2:{ n(z)}n∈Nis bounded},
K−={z∈C2:{ −n(z)}n∈Nis bounded},
K=K+∩K−,
U+=C2−K+,
U−=C2−K−,
and he G een unc ions:
G+(z, w) = lim
n→+∞
1
dnlog+| n(z, w)|,
G−(z, w) = lim
n→+∞
1
dnlog+| −n(z, w)|,
whe e d= deg( ) = deg( −1),
GK(z, w) = sup G+(z, w), G−(z, w).
234 C. Bisi
P oposi ion 2.6. [1],[2]I is a egula polynomial au omo phism
o C2o algeb aic deg ee d≥2, hen
•G+and G−a e con inuous unc ions on C2and
K+={G+= 0},
K−={G−= 0}.
•G+and G−a e plu iha monic (p.h.) espec i ely on U+and U−,
and plu isubha monic (p.s.h.) on C2.
•G+◦ =d·G+and G−◦ −1=d·G−.
•The closu e K+and K−o K+and K−in P2 e i y:
K+=K+∪I+,
K−=K−∪I−.
•I+is an a ac i e poin o −1and I−is an a ac i e poin
o .
•K=K+∩K−is a compac subse o C2.
P oo o Theo em 1.5: Fi s o all we wan o p o e ha :
(i) I dm
2≤dn
1wi h n, m ∈N, hen dm
2di ides dn
1.
Then we will p o e ha :
(ii) I o m, n ∈N,dm
2≤dn
1implies ha dm
2di ides dn
1, hen he e
exis n0, m0∈Nsuch ha dn0
1=dm0
2.
A i s way o p o e (i) is o p o e ha he G een unc ions o he
wo commu ing egula au omo phisms a e equal. The G een unc ion’s
app oach ex ends o Ck,k≥3.
Le G+
and G+
g he G een unc ions associa ed o and g. Conside
he unc ion:
H1=G+
◦g
d2
.
H1is a solu ion o he ollowing equa ion, because commu es wi h g:
(2.6) H1◦ =d1◦H1.
Hence H1and G+
sa is y he same unc ional equa ion.
Bu om [11], G+
is he la ges solu ion o he equa ion (2.6) among
he p.s.h. unc ions bounded by log+|z|+O(1) a in ini y.
Hence
H1=G+
◦g
d2
≤G+
On Comm. Polyn. Au om. o C2235
and also, ∀n∈N,
Hn=G+
◦gn
dn
2
≤G+
(2.7)
because G+
◦gn
dn
2
also sol es he equa ion (2.6).
On he o he hand, limn→+∞
G+
(gn)
dn
2=G+
g.
Indeed, i (z, w)∈K+
g, hen gn(z, w) is bounded when n→+∞; by
he con inui y o G+
we ha e ha G+
◦gn(z, w) is also bounded when
n→+∞, and hence
lim
n→+∞
G+
◦gn
dn
2
= 0 on K+
g.
Hence any limi unc ion o G+
(gn)
dn
2
is equal o G+
gon K+
g={G+
g= 0}.
On he o he hand, i (z, w)∈U+
g=C2−K+
gand (z, w) is in a
neighbo hood o I+
g,
log+|z|+c2≤G+
(z, w)≤log+|(z, w)|+c1
we ge
log+|z◦gn(z, w)|+c2
dn
2
≤G+
◦gn
dn
2
≤log+|gn(z, w)|+c1
dn
2
.
The i s and he las membe o he sequence o inequali ies end o
G+
g(z, w). The e o e limn→+∞
G+
(gn)
dn
2=G+
ge e ywhe e on C2. Hence
we ha e ha G+
g≤G+
and in e changing and gwe ge ha G+
g=
G+
=G+.
Obse e ha i would be su icien ha n◦gm=gm◦ n, o some
n, m > 1, in o de o ha e G+
=G+
g.
I ollows ha :
(2.8) G+( n◦g−m) = dn
1G+(g−m) = dn
1
dm
2
G+.
Suppose ha d2≤d1(i his is no he case, we ha e ha d1< d2
and we can use he same a gumen exchanging gwi h ) and conside
h:= g−1◦ .
The map his a polynomial au omo phism o C2which commu es wi h
a egula one ( o example o g), hence, by Co olla y 2.4, his a ine o
egula .
