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ONE-SIDED LEBESGUE BERNOULLI MAPS OF THE SPHERE
OF DEGREE
n
2AND
2n
2JULIA A. BARNES and LORELEI KOSS
(Received 24 November 1997 and in revised form 10 February 1998)
Abstract.We prove that there are families of rational maps of the sphere of degreen2
(n=2,3,4,...)and 2n2(n=1,2,3,...)which, with respect to a finite invariant measure equivalent to the surface area measure, are isomorphic to one-sided Bernoulli shifts of maximal entropy. The maps in question were constructed by Böettcher (1903–1904) and independently by Lattès (1919). They were the first examples of maps with Julia set equal to the whole sphere.
Keywords and phrases. One-sided Bernoulli, rational maps of the sphere, Julia sets.
2000 Mathematics Subject Classification. Primary 28D99.
1. Introduction. We consider rational maps of the sphere with Julia set equal to the whole sphere. Böettcher [3, 4] and Lattès [7] independently constructed the first exam-ples of such maps. A map is said to be one-sided Bernoulli of maximal entropy if it can be modeled by a faird-sided die toss, wheredis the degree of the map. Freire et al. [5] and independently Lyubich [8] conjectured that all rational maps of the sphere of degree greater than 1 are one-sided Bernoulli with respect to a particular measure, namely the measure of maximal entropy. Mañé [9] and Lyubich [8] proved that, for ev-ery rational map of the sphere of degree greater than 1, the maximal entropy measure is unique. Mañé [10] showed that for every rational map, there is some forward iterate which is one-sided Bernoulli with respect to the maximal entropy measure. However, the conjecture that all rational maps are one-sided Bernoulli is yet unknown. In [12], Parry discusses some of the reasons why proving that a map is one-sided Bernoulli is particularly difficult.
In this paper, we examine two families of rational maps which were constructed by Böettcher and Lattès, and we construct direct isomorphisms between these families of maps and one-sided Bernoulli shifts with respect to the maximal entropy measure. For the maps we examine here, the maximal entropy measure is equivalent to one of the most natural measures on the sphere, the normalized surface area measure. Furthermore, Zdunik [16] proved that the only maps that could possibly be one-sided Bernoulli with respect to a measure equivalent to the surface area measure are maps constructed in the fashion discussed here, i.e., as factors of toral endomorphisms.
property is introduced and the method used to prove Rω is one-sided Bernoulli is presented. Then Sections 5 and 6 are devoted to proofs concerning the two families of maps,RnandRn+ni, respectively, using an integral lattice. In the last section, we
extend the results from Sections 5 and 6 to hold for any appropriate lattice.
2. Basic construction. LetC∞denote the complex sphere andmdenote the
nor-malized surface area measure ofC∞. Letα,β∈C−{0}be such thatα/βis not a real
number. We define a lattice of points in the complex plane byΓ=[α,β]= {αi+βj:
i,j ∈ Z}. Given a lattice Γ and a complex number ω satisfying ωΓ ⊂ Γ, the map
Tω(z)=ωz (modΓ)gives an endomorphism of the complex torusC/Γ. In this paper, we restrict to mapsTω, whereω=n, n∈Z−{0,1,−1}, orω=n+ni, n∈Z−{0}.
Let℘Γ denote the Weierstrass elliptic function defined on a latticeΓ by
℘Γ(z)=z12+
ζ∈Γ−{0}
1
(z+ζ)2− 1
ζ2
. (2.1)
It is easy to see that℘Γis an even, onto, 2-fold branched cover ofC∞. Until Section 7
of this paper, we useΓ =[1,i], and we denote℘=℘[1,i]. We define a mapRω:C∞→C∞
byRω=℘Tω℘−1. The mapRωis well defined sinceTωΓ⊂Γ, and hence the following diagram commutes:
C/Γ
℘
Tω
/
/
C/Γ
℘
C∞ Rω
/
/
C∞.(2.2)
SinceRωis locally a composition of analytic maps,Rωis analytic. Further, all analytic maps ofC∞are rational, soRωis a rational map with degreed=ωω. Thus, forω=n, we have the deg(Rω)=n2and forω=n+ni, deg(Rω)=2n2.
It can be shown explicitly via algebraic identities thatR2(z)=(z2+1)2/[4z(z2−1)] as done in [2]. Also, Ueda [15] has proved that the rational mapR1+iof the sphere is given by the equationR1+i(z)=(1/2i)(z−(1/z)), a degree 2 map.
We represent the torus C/Γ by a period parallelogram Ω0= {x+iy :x,y ∈ R,
0≤x, y <1}, the unit square without the top or right boundaries. LetᏮ0be the Lebesgue measurable sets inΩ0, and letleb be normalized 2-dimensional Lebesgue measure restricted toΩ0. We use theσ-algebraᏮand measureνonC∞induced by the
factor map℘. ThusᏮis precisely theσ-algebra of Borel sets onC∞and the measure
ν is given byν(A)=leb(℘|−1
Ω0A) for allA∈Ꮾ. By definition, sinceTω is measure-preserving with respect toleb,Rω is measure-preserving with respect toν. We note that the measureνis equivalent tom, the normalized surface area measure onC∞.
Since℘ is an even map,℘(u)=℘(−u) for allu∈C. Therefore, the mapRω has critical values at precisely the points ℘(u) where u≡ −u (modΓ), as Rω fails to haveddistinct preimages at these points. There are exactly 4 points in the period parallelogramΩ0whereu≡ −u (modΓ). These are
V= 1
2,
i
2, 1+i
2 ,0
Therefore, the critical values (i.e., the images underRωof the critical points) of the rational mapRωare
e1=℘ 1
2
, e2=℘ i
2
, e3=℘ 1+i
2
, ℘(0)= ∞. (2.4)
The dynamical behavior of a rational mapRωdepends on the postcritical set ofRω, which is defined as
P(Rω)=
j>0
Rjω(c):cis a critical point ofRω
. (2.5)
Forω=norω=n+ni, P(Rω)= {e1,e2,e3,∞}. The postcritical orbit of the mapRω depends on whethernis even or odd.
For anyω=norω=n+niwithneven,
e1 → ∞ → ∞···, e2 → ∞ → ∞···, e3 → ∞ → ∞···. (2.6) Ifω=nandnis odd, then
e1 →e1···, e2 →e2···, e3 →e3···, ∞ → ∞···. (2.7) Ifω=n+niandnis odd, then
e1 →e3 → ∞ → ∞···, e2 →e3 → ∞→ ∞···. (2.8) In all cases,Rωhas a finite postcritical set, the critical values are periodic or preperi-odic, and none of the critical points ofRωare periodic (i.e., there are no critical points
c ofRω and no integersr >0 such that Rrω(c)=c). Therefore,Rω is ergodic and conservative (see [14]) as well as exact (see [1]) with respect tom.
3. Realization ofRω on the torus. In order to study the behavior of Rω on the sphere, we analyze the behavior of an isomorphic map on a subsetΩωof the topolog-ical closure of the period parallelogramΩ0as defined below. Due to the differences in the structure ofP(Rω)forneven andnodd, we defineΩωseparately forneven andnodd as follows:
Ifnis even,Ωω=
x+iy∈Ω0: 0≤y≤12
.
Ifnis odd,Ωω=x+iy∈Ω0:y≤x. (3.1)
See Figure 3.1 for an illustration ofΩω.
We next discuss some properties associated withΩω. For allu∈Ω0, we define an operation prime()byu=1+i−u∈Ω0. Note thatu= −u (modΓ)andu=u.
Lemma3.1. There is a set of leb-measure one inΩ0on which the following property holds:
u∈Ωωif and only ifu∈Ω0−Ωω. (3.2)
Proof. Let∂Ωωdenote the boundary ofΩω, andΩ0
0 1 i/2
0 1
1+i
neven nodd
Figure3.1. Illustration ofΩω.
points inΩ0
ω. Supposeu∈Ω0ω. Then the midpoint of the line segment connectingu anduinCis(u+u)/2=(1+i)/2 which is the center ofΩ0. Thusu∈Ω0−Ωωby
the symmetry ofΩ0.
The other direction works similarly.
Letθω=℘|Ωω. We recall that℘is a two-to-one (except at 4 points) surjective map,
and℘(u)=℘(u). From Lemma 3.1,θ
ω is measure-theoretically one-to-one. There-fore,θ−1
ω(z)has a unique value for everyz∈C∞in a set ofν-measure 1.
DefineTω:Ωω→Ωω such thatTω(u)=θω−1Rωθω(u)and the following diagram commutes:
Ωω
θω
Tω
/
/
Ωω
θω
C∞ Rω
/
/
C∞.(3.3)
LetᏮω be the Lebesgue measurable sets inΩω. We defineµ(A)=2leb(A)for all
A∈Ꮾω. Thus,Tωis well defined on a set ofµ-measure 1 andTωis measure preserving with respect toµ. Using the construction defined above, we are able to prove the following proposition which enables us to study the behavior ofRωby analyzingTω onΩω.
Proposition3.2. We have(Ωω,Ꮾω,µ,Tω)(C∞,Ꮾ,ν,Rω).
Proof. LetUbe a set ofleb-measure 1 inΩ0given in Lemma 3.1, and defineUω=
U∩Ωω. Thenθωis one-to-one onUω. By the definitions ofνandµ, 1=ν(℘(Uω))=
µ(θ−1
ω(℘(Uω)))=µ(Ωω)=ν(C∞).
Combining these facts with the construction of Tω above, we have shown that
θω:Uω→℘(Uω)is a measure-theoretic isomorphism, proving the proposition.
4. The one-sided Lebesgue Bernoulli property. LetX+
d = ∞
j=0{0,1,2,...,d−1}be the collection of one-sided sequences ondsymbols. LetᏯbe theσ-algebra generated by cylinder sets ofX+
d and letσbe the one-sided shift onXd+given byσ (x0,x1,x2,...)=
(x1,x2,...). Letpdbe the(1/d,1/d,...,1/d)Bernoulli measure onXd+, i.e.,pd([j0,j1,...,
We say that a map Rω is one-sided Bernoulli with respect to the measure ν if
(C∞,Ꮾ,ν,Rω)is isomorphic to (Xd+,Ꮿ,pd,σ ). Due to Proposition 3.2, this is equiv-alent to satisfying(Ωω,Ꮾω,µ,Tω)(Xd+,Ꮿ,pd,σ ).
To prove thatTωis one-sided Bernoulli with respect toµ, we construct a partitionᏼω of the spaceΩω intodcongruent atoms:A0,A1,...,Ad−1. Then we code the system
using a map ϕω :Ωω →Xd+ such that ϕω(u)=ci1,ci2,ci3,... whereT j
ω(u)∈Aij.
Notice thatTωj(Tω(u))=Tωj+1(u)∈Acj+1. Using this property, it is easy to show that σ ϕω(u)=ϕω(Tω(u)).
The mapϕωis the standard method used to code a system from a partition. The join of two partitionsηandξis defined to beη∨ξ= {Aj∩Bk:Aj∈η, Bk∈ξ}. We define
ᏼjω=jr=0Tω−r(ᏼω)=ᏼω∨Tω−1(ᏼω)∨ ··· ∨Tω−j(ᏼω), and we use Proposition 4.1 to show that the mapϕωis an isomorphism. A proof of Proposition 4.1 can be found in Petersen [13]; see Barnes [1] for the noninvertible case.
Proposition4.1. Ifᏼω generates underTω, then the map ϕω:Ωω→Xd+ is an
isomorphism.
To show thatᏼωis a generating partition, i.e.,∞j=0ᏼjωis the Borelσ-algebra(mod0), we apply Theorem 4.2, whose proof can be found in Mañé [11].
Theorem4.2. Let(X,Ᏸ,µ)be a probability space, whereX is a locally complete separable metric space andᏰis the Borelσ-algebra ofX. Letᏼjω, j≥1be a sequence
of partitions such that
lim j→∞
sup
P∈ᏼjω
diam(P)
=0. (4.1)
Then∞j=0ᏼjω=Ᏸ(mod0).
5. The familyRn. Recall thatT2(u)=2u (modΓ)andR2(z)=℘T2℘−1(z). We prove the following forR2.
Theorem5.1. ForΓ=[1,i], the rational mapR2(z)with measureν onC∞is
iso-morphic to the one-sided(1/4,1/4,1/4,1/4)Bernoulli shift on4symbols with measure p4; i.e.,(C∞,Ꮾ,ν,R2)(X4+,Ꮿ,p4,σ ).
Proof. In order to use the methods described in Section 4, we define a partition,
ᏼ2ofΩ2byᏼ2= {Ak: 0≤k≤3}, where
A0=
x+iy∈Ω2:x,y∈R,0≤x <12,0≤y <14
,
A1=
x+iy∈Ω2:x,y∈R, 1
2≤x≤1,0≤y < 1 4
,
A2=
x+iy∈Ω2:x,y∈R,0≤x <12, 14≤y≤12
,
A3=
x+iy∈Ω2:x,y∈R, 12≤x≤1, 14≤y≤12
.
To determine the partitionsᏼj2, we analyze images of the boundaries of the atoms ofᏼ2underT2−1. All of these boundaries are horizontal or vertical lines. Notice that any vertical line inΩ2 can be expressed by lv,k = {u∈Ω2: Re(u)=k}, for some real number k. T−1
2 (lv,k) is a pair of vertical lines described by {u∈Ω2: Re(u)=
k/2 or Re(u)=(k+1)/2}. Similarly, any horizontal line inΩ2is of the formlh,k= {u∈Ω2: Im(u)=k}for somek∈Rand also has 2 lines in its image underT−1
2 . One of these lines is the horizontal line satisfyingl1= {u∈Ω2: Im(u)=k/2}. The other line isl2= {u∈Ω2: Im(u)=(1−k)/2}. This comes from the fact that 2l2=Im(u)= 1−k∈Ω0−Ω2, and for allu∈ {z: Im(z)=1−k}, u∈ {z: Re(z)=k}. In Figure 5.1,
we illustrate the partitionsᏼ2andᏼ12.
0 1
i/2
0 1
i/2
A0
A2
A1
A3
A00
A02
A20
A22
A01
A03
A21
A23
A10
A12
A30
A32
A11
A13
A31
A33
ᏼ2 ᏼ12
Figure5.1. Partitions forT2.
Asj approaches∞, the sequence of partitions consists of rectangles with smaller and smaller diameter. We can calculate
lim j→∞
sup
P∈ᏼj2
diam(P)
=lim
j→∞
√ 5
24+2j=0. (5.2)
Therefore, by Proposition 4.1 and Theorem 4.2,ϕ2is an isomorphism. We extend the same type of argument to handle alln∈Z−{0,1,−1}.
Theorem5.2. ForΓ =[1,i], the rational mapRn(z), n∈Z− {0,1,−1}with
mea-sureνonC∞is isomorphic to the one-sided(1/n2,...,1/n2)Bernoulli shift onn2
sym-bols with measurepn2; i.e.,(C∞,Ꮾ,ν,R2)(X+n2,Ꮿ,pn2,σ ).
Proof. First we consider the case wherenis even. By increasing the value of the even number n, the partition becomes an n×n grid instead of a 2×2 grid. More precisely,ᏼn= {Ak: 0≤k≤n2−1}, where
Ak=
x+iy∈Ωn:k(modn n)≤x <k(modnn)+1, (k/n)−(k/n)(mod1)
2n ≤y≤
(k/n)−(k/n)(mod1)+1 2n
.
(5.3)
lim j→∞
sup
P∈ᏼjn
diam(P)
<lim
j→∞
√ 5
24+2j=0, (5.4)
andϕnis an isomorphism whennis even.
Now supposenis odd. ThenΩnis a triangle instead of a rectangle, but it still has half theleb-measure of the period parallelogramΩ0. As in the even case, the key to the proof comes from defining the appropriate partitionᏼn.
We defineᏼn= {Ak: 0≤k≤n2−1}, wheresk=(k/n)−(k/n)(mod1)and
Ak=
x+iy∈Ωnor(x+iy)∈Ωn:k(modn)
n ≤x <
k(modn)+1
n ;
y−x <sk−2[k(2nmodn)], y >2snk ifskis even;
y−x >sk−1−2[k(modn)]
2n , y < sk+1
2n ifskis odd
.
(5.5)
See Figure 5.2 for illustration ofᏼ3.
0 1
1+i
A0 A1 A2
A7 A8
A3
A4 A5
A6
Figure5.2. Partitionᏼ3.
This partition consists of right triangles which are contained in (1/n)×(1/n)
squares. Furthermore, we can calculate
lim j→∞
sup
P∈ᏼjn
diam(P)
=lim
j→∞
√ 2
nj+1=0, (5.6)
andϕnis an isomorphism.
6. The familyRn+ni. We next extend Theorem 5.2 to the family of rational maps
of the sphereRn+ni(z)=℘◦Tn+ni◦℘−1(z), whereTn+ni(u)=(n+ni)(u) (modΓ). Theorem6.1. LetΓ = [1,i]. The rational map Rn+ni(z) with measureν on C∞
is isomorphic to the one-sided(1/d,...,1/d)Bernoulli shift ond=2n2symbols with
measurepd; i.e.,(C∞,Ꮾ,ν,Rn+ni)(Xd+,Ꮿ,pd,σ ).
as follows:
A0=x+iy∈Ω1+i: x,y∈R, y≤x, y >−x+1,
A1=x+iy∈Ω1+i: x,y∈R, y≤x, y≤ −x+1.
(6.1)
Foru=a+bi∈C, we haveT1+i(u)=T1+i(a+ib)=(a−b)+(a+b)i. Thus the action ofT1+ion a pointuis to rotate it byπ/4 and increase its magnitude by√2, and
T−1
1+irotates the point by−π/4 and shrinks its magnitude by √
2. We show illustrations ofᏼ1+i,ᏼ11+i, andᏼ21+iin Figure 6.1.
0 1
1+i
A1
A0
0 1
1+i
A10 A11
A01
A00
0 1
1+i
A100
A101A111
A110
A011
A010
A000
A001
ᏼ1+i ᏼ11+i ᏼ21+i
Figure6.1.Partitions forT1+i.
The partitionjr=0T1−+ri(ᏼ1+i)cuts everyP∈ᏼj1−+1iinto two congruent triangles. Fur-ther, we can calculate for anyP∈ᏼj1+ithe diam(P)=1/√2j. Therefore,
lim j→∞
sup
P∈ᏼj1+i
diam(P)
=lim
j→∞
1 √
2j=0. (6.2)
Applying Proposition 4.1 and Theorem 4.2,ϕ1+iis an isomorphism, and we have com-pleted the proof in the casen=1.
Notice that the compositionTn◦T1+i=Tn+nigives a degree 2n2map of the torus. By using the above arguments and Theorem 5.2, we see thatRn+niis isomorphic to the one-sided(1/2n2,...,1/2n2)Bernoulli shift on 2n2symbols.
7. Changing the lattice. We prove that the above isomorphism extends to other tori C/Γ, whereTωΓ⊂Γ. SupposeΓ=[γ,δ]. Then we can find real numbersx
1,y1,x2, andy2such that
γ=x1α+y1β, δ=x2α+y2β. (7.1) The period parallelogram forC/ΓbecomesΩ
0= {cγ+dδ∈C: 0≤c , d <1}. We define a linear mapL:Ω0→Ω
0by
L(u)=L(a+ib)=
x1 y1
x2 y2
a b
We useLto define a measure-theoretic isomorphism between(Ωω,Ꮾω,µ,Tω)and
(Yω,ᏮY,µY,Tω,L), whereYω=L(Ωω)⊂Ω0,ᏮY is the Borel sets onYω,µY is normal-ized Lebesgue measure inYω, andTω,L=LTωL−1. We state this result in the following lemma.
Lemma7.1. We have(Ωω,Ꮾω,µ,Tω)(Yω,ᏮY,µY,Tω,Y). Lemma 7.1 provides us with the following corollary.
Corollary7.2. IfTω:C/Γ →C/Γ is given byTω(u)=ωu (modΓ), whereω=
n, n∈Z− {0,1,−1}orω=n+ni, n∈Z− {0}, then the rational mapRω(z)=℘Γ◦ Tω◦℘−Γ1(z)with measureνonC∞is isomorphic to the one-sided(1/d,...,1/d)Bernoulli
shift ond=ωωsymbols with measurepd; i.e.,(C∞,Ꮾω,ν,Rω)(Xd+,Ꮿ,pd,σ ). This technique is further developed by Koss [6] to analyze other classes of critically finite rational maps with parabolic orbifolds.
Acknowledgements. The authors wish to thank their advisor Professor Jane M. Hawkins for her guidance and support. This work was partially supported by GAANN Fellowships (Graduate Assistance in Areas of National Need) and NSF Grant DMS 9203489.
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Barnes: Department of Mathematics and Computer Science, Western Carolina Uni-versity, Cullowhee, NC28723, USA
E-mail address:barnes@wpoff.wcu.edu
Koss: Department of Mathematics, CB#3250, University of North Carolina at Chapel Hill, Chapel Hill, NC27599-3250, USA