Publ. Ma . 49 (2005), 127–157
A FAMILY OF CRITICALLY FINITE MAPS WITH
SYMMETRY
Sco C ass
Abs ac
The symme ic g oup Snac s as a e lec ion g oup on CPn−2
( o n≥3) . Associa ed wi h each o he `n
2´ ansposi ions in Sn
is an in olu ion on CPn−2 ha poin wise ixes a hype plane — he
mi o s o he ac ion. Fo each such ac ion, he e is a unique
Sn-symme ic holomo phic map o deg ee n+ 1 whose c i ical
se is p ecisely he collec ion o hype planes. Since he map p e-
se es each e lec ing hype plane, he membe s o his amily a e
c i ically- ini e in a e y s ong sense. Conside a ions o symme-
y and c i ical- ini eness p oduce global dynamical esul s: each
map’s Fa ou se consis s o a special ini e se o supe a ac ing
poin s whose basins a e dense.
1. O e iew
Complex dynamics in se e al dimensions has been he objec o con-
side able ecen s udy. Some specialized p e ious wo k in his ield ea s
a a ie y o maps ha sha e a common p ope y: hey espec he ac-
ion o a ini e g oup on a complex p ojec i e space. (See [C1], [C2],
[C3].) The na u e o hese in es iga ions leads o a conside a ion o
issues pe aining o global dynamics. While he mos signi ican dy-
namical claims possess expe imen al suppo , hey emain heo e ical
conjec u es. The cu en p ojec s ems om a desi e o ind symme -
ical maps wi h in e es ing geome y and ac able dynamics. I s i s
ui is an in ini e amily o special maps each o whose membe s espec
he ac ion o he symme ic g oup Sn. In ac , o each n≥3, he e
is a unique holomo phic map gon CPn−2whose c i ical se consis s o
an Sno bi o n
2hype planes ha gp ese es. This leads o a s ong
o m o c i ical ini eness ha yields se e al global dynamical esul s o
he ype ha eluded ea lie unde akings.
2000 Ma hema ics Subjec Classi ica ion. P ima y: 37F45; Seconda y: 20C30.
Key wo ds. Complex dynamics, equi a ian map, e lec ion g oup.
128 S. C ass
The ea men de elops in h ee s ages:
(1) Some backg ound on special ac ions o Snand hei associa ed
symme ical maps.
(2) P oo s ha he special amily o c i ically- ini e maps wi h Snsym-
me y exis s and ha each membe is unique and holomo phic.
(3) P oo s o claims conce ning he dynamics o he maps (in he
cases n= 3,4). Speci ically, each membe has a ce ain a ac-
o wi h dense basins. When n > 4, he claim conce ning he
a ac o is conjec u ed.
Finally, some g aphical esul s o low-dimensional cases appea .
2. Snac s on CPn−2
The pe mu a ion ac ion o he symme ic g oup Snon Cnp ese es
he hype plane
H=(n
X
k=1
xk= 0)≃Cn−1
and, he eby, es ic s o a ai h ul (n−1)-dimensional i educible ep-
esen a ion. This ac ion on Cn−1p ojec s one- o-one o a g oup Gn
on H:= PH ≃ CPn−2.
2.1. Special o bi s and e lec ion hype planes. The smalles Gno -
bi consis s o he npoin s
[1 −n, 1,...,1],...,[1,...,1,1−n].
(Squa e b acke s indica e poin s in p ojec i e space.)
Co esponding o he n
2 ansposi ions (ij) in Sna e n
2in olu ions
xi←→ xj
on H ha gene a e Gnas a complex e lec ion g oup. Each gene a ing
in olu ion ixes he poin
[0,...,0,
i
z}|{
1,0,...,0,
j
z}|{
−1,0,...,0]
and poin wise ixes he companion hype plane {xi=xj}.This poin -
hype plane pai gi es he only ixed poin s o he in olu ion. They o m
Gno bi s o size n
2. Fo ease o e e ence, use he e m “n
2-hype -
plane”.
C i ically Fini e Maps wi h Symme y 129
2.2. Coo dina es. The ans o ma ion A:Cn→Cn−1gi en by
u=Ax, A =




1 0 ... 0−1
0 1 ... 0−1
.
.
..
.
..
.
..
.
..
.
.
0 0 ... 1−1




=aij, aij =




1i=j
−1j=n
0 o he wise
gi es a special sys em o n−1 coo dina es on Hwhe e he n-poin o bi
is
[1,0,...,0],...,[0,...,0,1],[1,...,1].
No e ha he null space o Ais he euclidean o hogonal complemen
o H. This change o coo dina es has an “in e se”
x=Bu, B =






1−n1 1 ... 1 1
1 1 −n1... 1 1
.
.
..
.
..
.
..
.
..
.
..
.
.
1 1 1 ... 1 1 −n
1 1 1 ... 1 1







which gi es
AB =−n In−1, BA =1n−n In
whe e Imis he m×miden i y and 1nis he n×nma ix each en y
o which is 1. Acco dingly, Aand Binduce isomo phisms be ween H
and CPn−2.
In u-coo dina es, he n
2-hype planes a e he n−1 coo dina e hype -
planes {uk= 0}and he n−1
2spaces {uk=u`}.The poin s de e mined
by he in e sec ions o he n
2-hype planes play a cen al ole in subse-
quen de elopmen s. Thei desc ip ion is especially simple in u. (See
Table 1.) Wi h one excep ion, each o bi consis s o poin s pkand qk
wi h complemen a y coo dina es.
Rela i e o he uspace, Gnis gene a ed o e he pe mu a ion ac-
ion Gn−1o Sn−1on he ukby means o he in olu ion
T=






−100... 0 0
−110... 0 0
.
.
..
.
..
.
..
.
..
.
..
.
.
−100... 1 0
−100... 0 1







ha ansposes he pai {p1, q1}and ixes he emaining membe s o he
n-poin o bi . No e ha Tis he u e sion o he ans o ma ion
x1←→ xn.
130 S. C ass
Rep esen a i e poin s
non n−2 hype planes O bi size
2m−1pk= [
k
z}| {
1,...,1,
n−k−1
z }| {
0,...,0] n−1
k+n−1
k−1=n
k
qk= [
n−k
z}| {
1,...,1,
k−1
z }| {
0,...,0] k= 1,...,m−1
2m pk= [
k
z}| {
1,...,1,
n−k−1
z }| {
0,...,0] 

n
kk < m
1
2n
k=n−1
k−1k=m
qk= [
n−k
z}| {
1,...,1,
k−1
z }| {
0,...,0] k= 1,...,m
Table 1. Poin s de e mined by in e sec ions o n
2-hype planes.
3. Gnequi a ian s
Conside a map
= [ 1,..., n−1]
om H o i sel gi en by homogeneous polynomials in
u= (u1,...,un−1)
o deg ee . In gene al, can be me omo phic; ha is, o some p∈Cn−1,
(p) = 0 o e e y li o o Cn−1. We say ha is Gn-equi a ian when
i sends a g oup o bi o a g oup o bi . Algeb aically, his means ha
commu es wi h e e y elemen o Gn. Ob iously, is Gn−1-equi a ian
as well. I eadily ollows ha each componen kis in a ian unde he
s abilize Zko uk. Thus, we can exp ess a componen by
k=
X
`=0
u −`
kAk,`
whe e Ak,` is a deg ee-` Zkin a ian . Acco dingly, each Ak,` is aken o
be a polynomial in he elemen a y symme ic unc ions in he comple-
men a y a iables
buk= (u1,...,uk−1, uk+1,...,un−1).
Al e na i ely, we can employ he elemen a y symme ic unc ions in u
when exp essing Ak,`. This is a ma e o exp essing a polynomial in buk
C i ically Fini e Maps wi h Symme y 131
in e ms o a polynomial in uand a polynomial in bukwi h lowe de-
g ee. Speci ically, le b
Smand Smbe he deg ee-melemen a y symme ic
unc ions in bukand u espec i ely. Taking S0= 1, he ela ions
b
Sm=Sm−ukb
Sm−1
gi e a educ i e scheme o he eplacemen p ocess.
An immedia e consequence o Gn−1equi a iance is ha
Aj,` =Ak,` := A` o all j, k, `.
We can say a bi mo e conce ning he o m ha Gnequi a ian s ake.
Fi s , conside a poin a ha some elemen M∈ Gn ixes and obse e
ha
M (a) = (Ma) = (a).
Thus, ei he sends a o ano he ixed poin o Mo blows up a a
— ha is, o any li e
and eao and a o Cn−1,e
(ea) = 0. Applying
his condi ion o he n
2-hype planes, p o ided ha ais no a poin o
inde e minacy, each poin on such a hype plane mus map o a poin
ha is ixed by he in olu ion ha ixes he hype plane poin wise. The
only place o he image o such a poin is on he hype plane i sel o i s
companion poin . Unde a holomo phic map, he image canno be he
companion poin — his would o ce he en i e hype plane o collapse o
he poin . So, a holomo phic Gnequi a ian sends an n
2-hype plane
o i sel . This ci cums ance o ces k o be di isible by ukand, he eby,
equi es he e ms A o be a powe o Sn−1o o anish. In pa icula ,
when ≤n−1, A = 0 so ha
k=uk
−1
X
`=0
u −`−1
kA`.
By design, he map has Gn−1symme y. To be ully Gn-equi a ian ,
he map mus commu e wi h Tas well. This condi ion places s ong
es ic ions on he A`.The gene al o m hey ake migh be an in e es ing
esul , bu no one aken up by he cu en in es iga ion. He e, he ques
is o a amily o Gnequi a ian s wi h e y special p ope ies.
4. Re lec ion hype planes as c i ical se s: exis ence,
uniqueness, and holomo phy
Explici compu a ion in low-deg ee cases e eals he exis ence o a
unique holomo phic Gnequi a ian whose c i ical se is p ecisely he

132 S. C ass
n
2-hype planes coun ed wi h mul iplici y wo. These maps con o m o
a gene al o mula. Le
g= [g1,...,gn−1]
whe e
g`=u3
`G`, G`=
n−2
X
k=0
(−1)kk+ 1
k+ 3uk
`Sn,n−2−k,
and Sn,` is he deg ee-`elemen a y symme ic unc ion in u1,...,un−1.
In he deg ee-0 case, ake Sn,0= 1. By cons uc ion, each gis equi a ian
unde he g oup Gn−1 ha pe mu es he u`. In addi ion, he u3
` ac o
in each coo dina e implies ha he maps a e doubly c i ical on n−1 o
he n
2-hype planes —namely, whe e u`= 0. We e g o commu e wi h
he ans o ma ion T ha gene a es Gno e Gn−1, symme y would p o-
ide o double c i icali y on he emaining n−1
2o he n
2-hype planes
—whe e uj=uk.Mo eo e , since a deg ee-(n+1) map in n−1 a iables
has a c i ical se whose deg ee is
(n−1) n= 2n
2,
g’s c i ical se would consis exclusi ely o he n
2-hype planes.
This sec ion de elops a he echnical a gumen s o h ee main e-
sul s. Acco ding o Theo em 4.1, he n
2-hype planes o m g’s c i i-
cal se wi h mul iplici y wo. Mo eo e , Theo em 4.2 in o ms us ha
he e is only one such map o each Gnac ion. Theo em 4.3 s a es
ha each gis holomo phic on Hwhich implies ha gp ese es each
n
2-hype plane L a he han collapse L o a lowe -dimensional a ie y;
a con ac ion would o ce he map o blow up.
Thus, gis a amily o maps each membe o which is holomo phic,
doubly-c i ical on he n
2-hype planes, and c i ically- ini e. As a s and-
ing assump ion, le n≥3.
Theo em 4.1. The espec i e gis T-equi a ian , hence, Gn-equi a ian .
Theo em 4.2. Unde he ac ion o Gn,gis he unique a ional map o
deg ee n+ 1 o which each n
2-hype plane is doubly c i ical.
Theo em 4.3. Each membe o he amily gis holomo phic on H.
P oo o Theo em 4.1: P oposi ions 4.5 and 4.8 below es ablish ha g
is symme ic unde Tas well as unde Gn−1. Since Tgene a es Gn
o e Gn−1,gis Gn-equi a ian .
The p oo s o he p oposi ions ely on a o mula ha desc ibes how
he elemen a y symme ic unc ions ans o m unde T. This esul
C i ically Fini e Maps wi h Symme y 133
was ound by pa e n de ec ion in low-deg ee cases. Fo simplici y o
appea ance, exp ess he unc ions Sn,k(u) in he supp essed o m Sn,k.
Lemma 4.4. Fo k≤n, he Gn−1in a ian s Sn,k ans o m unde T
acco ding o
Sn,k(Tu) =
k
X
`=0
(−1)`n−k+`
n−ku`
1Sn,k−`.
P oo : P oo s o se e al echnical lemmas appea a [C4].
The a gumen o he T-equi a iance o gexamines he coo dina es
indi idually.
P oposi ion 4.5. The ac o G1o g1is T-in a ian (in a linea as
well as p ojec i e sense).
P oo : The p oo amoun s o manipula ion o sums. Since nis ixed
he e, le Sk=Sn,k. Conside
G1(Tu)=
n−2
X
k=0
(−1)kk+ 1
k+ 3(−u1)kSn−2−k(Tu)=
n−2
X
k=0
k+ 1
k+ 3uk
1Sn−2−k(Tu).
By Lemma 4.4,
G1(Tu) =
n−2
X
k=0
k+ 1
k+ 3uk
1 n−2−k
X
`=0
(−1)`n−(n−2−k) + `
n−(n−2−k)u`
1Sn−2−k−`!
=
n−2
X
k=0
k+ 1
k+ 3 n−2−k
X
`=0
(−1)`k+`+ 2
k+ 2 uk+`
1Sn−2−(k+`)!.
Se ing m=k+`,
G1(Tu) =
n−2
X
k=0
k+ 1
k+ 3 n−2
X
m=k
(−1)m−km+ 2
k+ 2 um
1Sn−2−m!.
Re e sing he o de o summa ion,
G1(Tu) =
n−2
X
m=0 m
X
k=0
(−1)m−kk+ 1
k+ 3m+ 2
k+ 2 !um
1Sn−2−m
=
n−2
X
m=0
(−1)m(m+ 2)! m
X
k=0
(−1)kk+ 1
(k+ 3)!
1
(m−k)!!um
1Sn−2−m.
134 S. C ass
Lemma 4.6 below gi es he sum o e k:
G1(Tu) =
n−2
X
m=0
(−1)m(m+ 2)! m+ 1
(m+ 3)!um
1Sn−2−m
=
n−2
X
m=0
(−1)mm+ 1
m+ 3um
1Sn−2−m
=G1(u).
Lemma 4.6.
m
X
k=0
(−1)kk+ 1
(k+ 3)!(m−k)! =m+ 1
(m+ 3)!.
P oo : See [C4].
Co olla y 4.7. Each gis T-equi a ian in he i s coo dina e.
P oo : Le [·]1speci y a map’s i s coo dina e. Then
g1◦T=−u3
1G1◦T=−u3
1G1= [T◦g]1.
To es ablish o e all T-equi a iance, i su ices o conside he beha io
o gunde Tin jus he second coo dina e. This ollows di ec ly om he
commu a i i y o Tand he membe s τ2,m ∈ Gn ha simply anspose
he second and m h basis elemen s:
[0,1,0,...,0] τ2,m
←→ [0,0,...,0,1
|{z}
m
,0,...,0]
p o ided ha m6= 1, n. Exp essed in e ms o Sn, his amoun s o he
commu a i i y o he disjoin ansposi ions (1n) and (2m). So, no ing
ha gis Gn−1-equi a ian , hence, τ2,m-equi a ian , and gi en ha gis
T-equi a ian in i s second coo dina e,
gm◦T=[g◦T]m
=[(τ2,m ◦g◦τ2,m)◦T]m=[τ2,m ◦(g◦T◦τ2,m)]m=[g◦T◦τ2,m]2
=[g◦T]2◦τ2,m = [T◦g]2◦τ2,m = [T◦g◦τ2,m]2
=[τ2,m ◦T◦g]2= [T◦g]m.
P oposi ion 4.8. The second coo dina e o gsa is ies he equi a iance
condi ion
g2◦T= [T◦g]2.
C i ically Fini e Maps wi h Symme y 135
P oo : Fi s , exp ess g2◦Tin a way ha ’s use ul o compa ison
o [T◦g]2. Again, se Sk=Sn,k. Applying Lemma 4.4,
g2(Tu)=(u2−u1)3
n−2
X
k=0
(−1)kk+ 1
k+ 3(u2−u1)kSn−2−k(Tu)
=
n−2
X
k=0
k+ 1
k+ 3 n−2−k
X
`=0
(−1)k+`
k+2+`
k+ 2 u`
1Sn−2−k−`!(u2−u1)k+3.
Se ing m=k+`,
g2(Tu) =
n−2
X
k=0
k+ 1
k+ 3 n−2
X
m=k
(−1)mm+ 2
k+ 2 um−k
1Sn−2−m!(u2−u1)k+3.
Re e sing he o de o summa ion,
g2(Tu) =
n−2
X
m=0
(−1)m m
X
k=0
k+ 1
k+ 3m+ 2
k+ 2 um−k
1(u2−u1)k+3!Sn−2−m.
Lemma 4.9 below es ablishes a use ul iden i y o he sum o e kso ha
g2(Tu) = u3
2
n−2
X
m=0
(−1)mm+ 1
m+ 3um
2Sn−2−m
−u3
1
n−2
X
m=0
(−1)mm+ 1
m+ 3um
1Sn−2−m
−u1u2
n−2
X
m=0
(−1)m(um+1
2−um+1
1)Sn−2−m.
The i s wo e ms a e g2(u) and g1(u) espec i ely. Since hei di e -
ence amoun s o [Tg(u)]2,
Tg(u)2−g2(Tu) = u1u2
n−2
X
m=0
(−1)m(um+1
2−um+1
1)Sn−2−m.
Adding and sub ac ing −u1u2Sn−1on he igh ,
Tg(u)2−g2(Tu) = u1u2 −Sn−1+
n−2
X
m=0
(−1)mum+1
2Sn−2−m
−−Sn−1+
n−2
X
m=0
(−1)mum+1
1Sn−2−m!.
142 S. C ass
Wi h his,
G1(pm) =
n−2
X
k=0
(−1)kk+ 1
k+ 3Sn,n−2−k(pm)
=
n−2
X
k=n−2−m
(−1)kk+ 1
k+ 3m
n−2−k.
Se ing p=n−2−k,
G1(pm) =
m
X
p=0
(−1)n−2−pn−p−1
n−p+ 1m
p
= (−1)n
m
X
p=0
(−1)pn−p−1
n−p+ 1m
p.
F om Lemma 4.13 below,
G1(pm) = (−1)n2 (−1)m−1
(n+ 1)n
m6= 0.
Lemma 4.13.
m
X
p=0
(−1)pn−p−1
n−p+ 1m
p=2 (−1)m−1
(n+ 1)n
m.
P oo : See [C4].
5. Re lec ion hype planes as c i ical se s: global
dynamics
Le Ln−3gene ically deno e an n
2-hype plane and le X e e o he
union o he Ln−3. Whe e mo he Ln−3in e sec o o m a CPn−2−m,
call he esul ing space Ln−2−m. (No e ha mo e han mo he Ln−3
can pass h ough an Ln−2−m.)
No only is gc i ically- ini e on H≃CPn−2wi h c i ical se consis -
ing o he Ln−3hype planes, he es ic ion gLn−2−mis also c i ically-
ini e, ha ing a collec ion o he Ln−3−m o i s c i ical se . In [FS1],
such beha io is called s ic c i ical ini eness (Sec ion 7). In ac , all
o he Ln−3−mon an Ln−2−ma e c i ical o gLn−2−m hough no wi h
he same mul iplici y.

C i ically Fini e Maps wi h Symme y 143
5.1. The Fa ou se o g.Following s anda d p ac ice, he Fa ou se Fg
is whe e he amily o i e a es {gk}is no mal and he Julia se Jgis he
complemen o Fg.
The beha io o gon an Ln−3plays a cen al dynamical ole. Again,
li g o Cn−1:
eg= (g1,...,gn−1)
wi h
g`=u3
`G`and G`=
n−2
X
k=0
(−1)kk+ 1
k+ 3uk
`Sn,n−2−k.
Fo a space Lm⊂CPkli ed o Ck+1, call he li ed space e
Lm+1.
P oposi ion 5.1. Fo any a∈ Ln−3,gis c i ical in he di ec ion o o
he hype plane.
P oo : By symme y, conside he e
Ln−2gi en by {u1= 0}. Fo any
a∈ {u1= 0}, he i s ow o Deg(a) anishes. Thus, he local beha io
o egcollapses poin s on o e
Ln−2. Explici calcula ion e eals ha he
collapse occu s in he di ec ion o (2,1,...,1).
Recall ha he pm ep esen he poin se s o Gno bi s de e mined
by in e sec ing he Ln−3. Re e o hese o bi s as “pm-poin s”. Fi s o
all, each such poin is supe a ac ing in all di ec ions.
Theo em 5.2. Unde g, he ixed pm-poin s a e supe a ac ing in e e y
di ec ion. Con e sely, he only poin s ha a e supe a ac ing in e e y
di ec ion a e he pm-poin s.
P oo : To es ablish ha , a pm,gis c i ical in e e y di ec ion in CPn−1
show ha he Jacobian Dega pmhas ank 1. He e, pmis li ed in he
li e al way. I hen ollows ha , since eg(pm)6= 0, he e a e n−2 non-
adial di ec ions h ough pm ha ha e ze o eigen alue.
The Jacobian has he o m
Deg=(aij) (bij)
0 0 
whe e
aij =




3Gi(pm) + ∂Gi
∂ui
(pm)i=j
∂Gi
∂uj
(pm)i6=j
, i, j ≤m
bij =∂Gi
∂um+j
(pm), i ≤m < j.
144 S. C ass
Wi h Sk=Sn,k a s aigh o wa d calcula ion es ablishes ha , o `≤m,
∂Sk
∂u`
(pm) = (0k > m
m−1
k−1k≤m
so ha ∂Gi
∂uj(pm) is he same alue o i, j ≤mwi h i6=j. Simila ly,
∂Gi
∂u`(pm) is he same alue o ` > m. I emains o show ha
3Gi(pm) + ∂Gi
∂ui
(pm) = ∂Gj
∂uk
(pm) o all i, j, k ≤m.
By manipula ion o sums,
∂Gi
∂ui
(pm) =
n−2
X
k=0
(−1)n−kn−k−1
n−k+ 1 (n−2−k)Sk(pm)
+
n−2
X
k=0
(−1)n−kn−k−1
n−k+ 1
∂Sp
∂ui
(pm).
The second sum is ∂Gj
∂u`(pm) o j, ` ≤mand j6=`. To show ha he i s
sum amoun s o −3Gi(pm), no ice ha , om he p oo o Theo em 4.3,
n−2
X
k=0
(−1)n−kn−k−1
n−k+ 1(n−2−k)Sk(pm)
= (−1)n
m
X
k=0
(−1)kn−k−1
n−k+ 2(n−2−k)m
k
= (−1)n
m
X
k=0
(−1)kn−k−1
n−k+ 1((n−k+ 1) −3)m
k
= (−1)n
m
X
k=0
(−1)k(n−1−k)m
k−3Gi(pm).
Finally, he calcula ion educes o showing ha he i s sum anishes.
This ollows eadily by spli ing he sum in o wo e ms each o which is
C i ically Fini e Maps wi h Symme y 145
a binomial expansion o 1 −1. Speci ically,
m
X
k=0
(−1)k(n−1−k)m
k= (n−1)
m
X
k=0
(−1)km
k−
m
X
k=0
(−1)kkm
k
= (n−1)(1 −1)m+m
m
X
k=1
(−1)km−1
k−1
=m(1 −1)m−1.
Thus, he nonze o ows o Deg(pm) a e iden ical and he ma ix has
ank 1.
Fo he con e se claim, conside a poin q ha is c i ical in e e y
di ec ion. When gis es ic ed o any in e sec ion Lko hype planes each
o which is an Ln−2,qis again c i ical o he es ic ion gLk. Hence,
qlies on some Ln−2 ha does no con ain Lkand so, is de e mined by
he in e sec ion o Ln−2spaces.
Now o he issue o he Fa ou se Fg. Is he e a Fa ou componen
o g ha is no in he basin o a pmpoin ?
Theo em 5.3. Fo n= 3,4,Fgconsis s o he basins o a ac ion o
he pm-poin s.
P oo : When n= 3, he one-dimensional map ghas h ee ixed c i ical
pm-poin s. A basic esul in one-dimensional dynamics s a es ha he
Fa ou se o a a ional map wi h pe iodic c i ical poin s consis s only o
supe a ac ing basins; indeed, he basins ha e ull measu e in CP1.
In he wo-dimensional case n= 4, he claim ollows om Theo em 5.2
and [FS1, Theo em 7.7]. The la e implies ha i a holomo phic map
on CP2has a c i ical se Csuch ha 1) Cis pe iodic and 2) CP2−C
is Kobayashi hype bolic, hen has only supe a ac ing basins in i s
Fa ou se . See below o an explana ion o he ac ha condi ion 2)
applies o g.
The gene al case emains open.
Conjec u e 5.4. Fo n≥5,Fgconsis s o he basins o a ac ion o
he pm-poin s.
One app oach o his claim adop s a echnique om he p oo o
Theo em 4.3: educ ion o dimension o he one-dimensional case whe e
some hings a e unde s ood. The a gumen o Theo em 5.6 employs
he same idea. Assume an a bi a y choice o n≥5.
146 S. C ass
The ques ion o whe he he basins o he pm-poin s exhaus Fgcalls
o some p epa a ion. Following [U], le C be he c i ical se o a
holomo phic map on CPm,
D :=
∞
[
k=1
k(C ) and E :=
∞
k=1
k(D )
be he pos c i ical se and he ω-limi se o C espec i ely. Also, he
Fa ou limi se Λ is whe e he o wa d o bi s o Fa ou componen s
accumula e. In he case o g,Dg=Eg=X.
Le p∈Fgand Ube he Fa ou componen o which pbelongs. Fo
a c i ically- ini e map , Λ ⊂E [U, Theo em 5.1]. Acco dingly, he
o wa d o bi {gk(p)}o paccumula es on some Ln−3and, by P oposi-
ion 5.1, is a ac ed o ha Ln−3—call i Ln−3as well. Acco dingly,
gn(U)−→ Ln−3.
The claim also ollows om [M, Theo em 2.36] —a esul es ablished by
conside a ion o expansion in he Kobayashi me ic on he complemen
o he pos c i ical se .
The ask now is o show ha
g (U)∩Ln−36=∅
o some . An a gumen migh de elop in wo s eps: 1) he o bi o
a poin ha is Fa ou o gaccumula es a poin s ha a e Fa ou o
eg:= gLn−3; 2) a poin ha is Fa ou o egis also Fa ou o gand,
he eby, belongs o a Fa ou componen in CPn−2.
To ea he i s claim, le q∈ Ln−3be a limi poin o {gk(p)}
wi h gnkK→hwhe e h:K→ Ln−3,K⊂Uis a neighbo hood o p,
and h(p) = q.
Suppose ha qbelongs o he Julia se Jeg. By P oposi ion 5.1, gis
supe a ac ing a gk(q) in some di ec ion away om Ln−3 o all k.
This equips qwi h a s able se
Sq={x|dis (gnk(x), gnk(q)) −→ 0}
ans e se o Ln−3. I egwe e hype bolic—as in he case n= 4, one
migh expec ha he Kobayashi expansion a qwould p oduce saddle-
like beha io and o ce U o con ain Julia poin s o g.
C i ically Fini e Maps wi h Symme y 147
To see claim 2) abo e, le q∈Fegwi h a neighbo hood e
Non which
{egk}is no mal. Take N o be he connec ed neighbo hood o q ha is
abso bed by e
Nand includes e
N; ha is, Nis he connec ed componen
o he s able se o e
N
Se
N=[
x∈
e
N
Sx
whe e e
N⊂Nand Sxis he s able se o x. E e y poin in Nbelongs o
some Sx. Thus, i egnkcon e ges o ehon e
N, hen gnkcon e ges on N o
h(y) = eh(x), y ∈Sx.
The claims 1) and 2) imply ha some g (U) in e sec s Ln−3; indeed,
g (U)∩Ln−3is a Fa ou componen o eg. By he c i ical ini eness o eg,
he o wa d o bi o g (U)∩Ln−3mee s some Ln−4in Fa ou poin s o
gLn−4.
This cascade con inues un il some gs(U) makes con ac wi h a line L1,
in pa icula , wi h he Fa ou se o gL1. Since gL1has ixed c i ical
poin s, i has only supe a ac ing basins. The only c i ical poin s on L1
a e pm-poin s. Hence, gs(U)∩L1lies in he basin o a ac ion o some
such poin .
How “la ge” a e he basins o he pm-poin s? Fi s o all, le
B := [
k≥0
−k(C )
be he p ec i ical se o . The ollowing basic esul yields ha he
closu e o Bgcon ains he Julia se Jg[FS1, P oposi ion 6.5].
Theo em 5.5. I :CPk→CPkis holomo phic and CPk−B is
hype bolically embedded,
J ⊂A :=
n>0[
m>n
−m(C ).
To apply his esul o g, we mus see ha i sa is ies he hypo he-
ses. By Theo em 4.3, gis holomo phic on CPn−2. Two heo ems o
M. G een imply ha Ln−1−m−BgLn−1−m
is hype bolically embedded
in Ln−1−m( aking Ln−2=H). (Fo de ails on G een’s esul s, consul
[FS1, Sec ion 5].) To see his, suppose ha , o n≥4 and m≥2,
φ:C−→ Ln−1−m−BgLn−1−m
is holomo phic. Then φ(C) omi s a leas n−m+ 1 hype su aces
in Ln−1−m, namely, some Ln−m−2spaces and hei p eimages. By one

148 S. C ass
o G een’s heo ems (Theo em 5.6 in [FS1]), φ(C) is con ained in a
compac complex hype su ace. Since such a hype su ace in e sec s he
omi ed hype su aces, φ(C) omi s a leas h ee poin s and so, is con-
s an . The s a emen conce ning hype bolic embedding ollows om
G een’s o he heo em (Theo em 5.5 in [FS1]).
One o he p elimina y: since Cg⊂g−1(Cg), Jg⊂Ag=Bg. We can
now es ablish a bi o Fg’s global s uc u e.
Theo em 5.6. Unde he assump ion ha Conjec u e 5.4 holds, he
Fa ou se Fgis dense in H.
P oo : Conside j0∈Jgand le U0be a neighbo hood o j0. By Theo-
em 5.5, some p ec i ical poin s mee U0so ha , o some m,
gm(U0)∩Cg6=∅.
I
U1:= gm(U0)∩Ln−3
ails o con ain Julia poin s, he case is made. O he wise, ake a Julia
poin j1∈U1, a neighbo hood o j1.
The map gLn−3is c i ically ini e wi h c i ical se Cn−3in he in e -
sec ion o Ln−3and he hype planes in Xdi e en om Ln−3. Hence,
Cn−3is a collec ion o Ln−4spaces. Implemen ing he a gumen gi en
o j0and U0unde gusing j1and U1unde gLn−3p oduces a neigh-
bo hood o a Julia poin j2on some Ln−4. The descen con inues un il
i eaches a Julia poin jn−3and neighbo hood Un−3on an L1. Thus,
Un−3mee s he Fa ou se o gL1. Since gL1has ixed c i ical poin s
ha a e pm-poin s, i s Fa ou se consis s o he supe a ac ing basins
o hose pm-poin s. Acco dingly, Un−3—hence, U0— con ains poin s
in Fg.
5.2. A que y on he s uc u e o g’s Julia se . Fo he es ic ed
map bg=gLn−3, he Julia se is gi en by
Jbg=Jg∩Ln−3.
The inclusion Jbg⊂Jg∩ Ln−3is clea . I x /∈Jbg, hen xbelongs o
a basin o a pm-poin so ha x /∈Jg. A each poin p∈Jbg, he map
is supe a ac ing in he di ec ion away om Ln−3. Thus, he e is a
“s able se ” Spo poin s in Jgwhose o bi s a e a ac ed o he o bi
o p. Acco dingly, he e is a s able bundle o e Jbg
SJbg:= [
p∈Jbg
Sp⊂Jg.
C i ically Fini e Maps wi h Symme y 149
A e he Spone-dimensional mani olds? A e he p eimages o he SJbg
dense in Jg?
In he case n= 4, g es ic s o a c i ically- ini e map bgon an L1 ha
is one o he six lines o e lec ion o G4. Figu e 4 displays he h ee
basins o a ac ion o bg. The Julia se Jbgconsis s o he bounda ies o
hese basins. Fo each Julia poin p∈ L1, he e is an Spaway om he
line. Wha can be said abou he s uc u e o SJbg?
Wha abou he poin s
K:= Jg−[
X
SJbg
ha a e no abso bed by X?
On an L1, each Julia poin is non-wande ing and has a con ac ing
di ec ion on o L1and an expanding di ec ion in L1. Fo a hype bolic
map on CP2, he li e a u e desc ibes a g ading o he non-wande ing
se Ω by he expanding dimension [FS2]:
Ω = Ω0∪Ω1∪Ω2.
The pm-poin s comp ise Ω0and ∪XJbg⊂Ω1. The non-wande ing poin s
no on Xbelong o K. Since any neighbo hood o such a poin pcon ains
an open se ha is a ac ed o X, he e is expansion a p. Does i
happen ha
Ω∩K⊂Ω2
so ha gis hype bolic?
6. Geome y and dynamics in low-dimension
To a oid con usion, le gn+1 ep esen he pa icula map gon he
espec i e Gn-symme ic H.
6.1. The one-dimensional case: g4and Halley’s me hod. When
n= 3, he e lec ing “hype planes” consis o a h ee-poin o bi . Wi h
hese poin s loca ed a
{1, ρ, ρ2|ρ=e2π i/3},
he map’s inhomogeneous exp ession on {u26= 0}is
z−→ z(z3−2)
2z3−1.
We can ealize he G3ac ion on CP1by he polyhed al con igu a ion
o a double iangula py amid — wo egula e ahed a joined a a ace.
The wo-poin o bi esides a 0 and ∞and de ines wo hemisphe es in
he usual way. Acco dingly, he uni ci cle co esponds o he equa o ial
150 S. C ass
bounda y be ween hemisphe es and he 3-poin s {1, ρ, ρ2}a e e ices
whe e ou aces cong ega e.
Conside he deg ee-4 map ha ixes he e ices o each ace and
sends one ace F o ou o he s: Fi sel and he h ee aces in he
hemisphe e no con aining F. This symme ical cons uc ion esul s in
G3-equi a ian beha io . A he h ee equa o ial e ices, he map opens
up a ace’s in e nal angle o π/2 o an angle o 3 π/2 so ha he local
beha io is cubing. This makes he 3-poin o bi doubly-c i ical and,
by deg ee coun ing, he en i e c i ical se . Acco dingly, his map mus
be g4. Since g4has pe iodic c i ical poin s, he supe a ac ing basins
cons i u e i s Fa ou se and, mo eo e , ha e ull measu e in CP1. A
po ai o he basins appea s in Figu e 1.
I u ns ou ha g4is Halley’s Me hod —a a ia ion on New on’s
Me hod— o a cubic polynomial. (See [ST] o a desc ip ion o Hal-
ley’s Me hod in eal a iables.) In he coo dina es selec ed abo e, he
polynomial o which we apply Halley’s me hod is
z3−1.
Figu e 1. Dynamics o g4on he S3-symme ic CP1
C i ically Fini e Maps wi h Symme y 151
6.2. The map in wo dimensions. Since gn+1 has eal coe icien s,
i p ese es he RPn−2o poin s whose coo dina es can be exp essed by
eal numbe s. Call his space R. Unde G4,Rhas he s uc u e o a
p ojec i e cube. We can iew his as a hemisphe e whe e one e ex is
a he pole and he o he h ee e ices lie along a ci cle whose cen e
is he dis inguished e ex. The 3-poin o bi (i.e., he ace-cen e s) lies
on ano he ci cle cen e ed a he no h pole.
Figu e 2 displays he basins o a ac ion o g5on R. In he a ine
plane o he pic u e, he e ices o he cube a e
(0,0),(1,0), −1
2,±√3
2!
while he h ee ace-cen e s a e he edge-midpoin s
−1
2,0, 1
4,±√3
4!
o he equila e al iangle o med by he h ee e ices ha a e no (0,0).
The map is gi en by
(x, y)−→
3`15 x4+ 12 x5−30 x2y2−5y4+ 20 x y4´
1−10 x2+ 20 x3+ 30 x4+ 40 x5−10 y2−60 x y2+ 60 x2y2−80 x3y2+ 30 y4−120 x y4,
24 `−5x y3+ 5 x2y3+y5´
1−10 x2+ 20 x3+ 30 x4+ 40 x5−10 y2−60 x y2+ 60 x2y2−80 x3y2+ 30 y4−120 x y4!.
The six lines o e lec ion un along he edges and a diagonal o a ace.
These lines ca e he hemisphe e in o wel e iangles each o which is
a undamen al domain o he e lec ion g oup ac ion G4. Viewing he
“hemi-cube” om abo e an edge, Figu e 3 e eals he map’s ac ion on
a undamen al iangle: one iangle s e ches and wis s on o i e o he
associa ed iangles.
Re u ning o ucoo dina es, one o he six mi o s —say, {u3= 0}—
is Z2-s able. Res ic ed o his line, g5has h ee supe a ac ing poin s:
•A wo-poin Z2o bi o ype p1poin s [1,0,0] and [0,1,0] (whe e
{u2= 0}and {u1= 0}in e sec {u3= 0}).
•A one-poin Z2o bi o he poin p2= [1,1,0] (whe e {u1=u2}
in e sec s {u3= 0}).