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©Hindawi Publishing Corp.

ONE-SIDED LEBESGUE BERNOULLI MAPS OF THE SPHERE

OF DEGREE

n

2

AND

2n

2

JULIA 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.

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property is introduced and the method used to prove 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. LetCdenote 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}.

LetΓ 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 thatΓis 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 mapis well defined sinceTωΓ⊂Γ, and hence the following diagram commutes:

C/Γ

/

/

C/Γ

C

/

/

C∞.

(2.2)

Sinceis locally a composition of analytic maps,is analytic. Further, all analytic maps ofCare rational, sois 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(z21)] 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 measureνonCinduced by the

factor map. ThusᏮis precisely theσ-algebra of Borel sets onCand the measure

ν is given byν(A)=leb(℘|−1

Ω0A) for allA∈Ꮾ. By definition, since is measure-preserving with respect toleb, 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 map has critical values at precisely the points ℘(u) where u≡ −u (modΓ), as 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

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Therefore, the critical values (i.e., the images underof the critical points) of the rational mapare

e1=℘ 1

2

, e2=℘ i

2

, e3=℘ 1+i

2

, ℘(0)= ∞. (2.4)

The dynamical behavior of a rational mapdepends on the postcritical set ofRω, which is defined as

P(Rω)=

j>0

Rjω(c):cis a critical point of

. (2.5)

Forω=norω=n+ni, P(Rω)= {e1,e2,e3,∞}. The postcritical orbit of the map 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,has a finite postcritical set, the critical values are periodic or preperi-odic, and none of the critical points ofare periodic (i.e., there are no critical points

c of and no integersr >0 such that Rrω(c)=c). Therefore, is ergodic and conservative (see [14]) as well as exact (see [1]) with respect tom.

3. Realization of on the torus. In order to study the behavior of 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+iuΩ0. 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 ifuΩ0Ωω. (3.2)

Proof. LetΩωdenote the boundary ofΩω, andΩ0

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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. ThusuΩ0Ωωby

the symmetry ofΩ0.

The other direction works similarly.

Letθω=℘|Ωω. We recall thatis 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∈Cin a set ofν-measure 1.

Define:ΩωΩω such thatTω(u)=θω−1Rωθω(u)and the following diagram commutes:

Ωω

θω

/

/

Ωω

θω

C

/

/

C∞.

(3.3)

LetᏮω be the Lebesgue measurable sets inΩω. We defineµ(A)=2leb(A)for all

A∈ω. Thus,Tωis well defined on a set ofµ-measure 1 andis 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 ofby analyzing onΩω.

Proposition3.2. We have(Ωω,ω,µ,Tω)(C∞,,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 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,...,

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We say that a map is one-sided Bernoulli with respect to the measure ν if

(C∞,,ν,Rω)is isomorphic to (Xd+,,pd,σ ). Due to Proposition 3.2, this is equiv-alent to satisfying(Ωω,ω,µ,Tω)(Xd+,,pd,σ ).

To prove thatis 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

jω=jr=0Tω−r(ω)=ω∨Tω−1(ω)∨ ··· ∨Tω−j(ω), 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. Ifω generates underTω, then the map ϕω:Ωω→Xd+ is an

isomorphism.

To show thatᏼωis a generating partition, i.e.,∞j=0ᏼis the Borelσ-algebra(mod0), we apply Theorem 4.2, whose proof can be found in Mañé [11].

Theorem4.2. Let(X,,µ)be a probability space, whereX is a locally complete separable metric space andis the Borelσ-algebra ofX. Letjω, j≥1be a sequence

of partitions such that

lim j→∞

  sup

P∈ᏼjω

diam(P)

 

=0. (4.1)

Then∞j=0ᏼjω=(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∞,,ν,R2)(X4+,,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

.

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To determine the partitionsᏼj2, we analyze images of the boundaries of the atoms ofᏼ2underT21. 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∈Ω0Ω2, 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∞,,ν,R2)(X+n2,,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≤n21}, 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)

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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≤n21}, 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−12[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∞,,ν,Rn+ni)(Xd+,,pd,σ ).

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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 by2, 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∈j1−+1iinto two congruent triangles. Fur-ther, we can calculate for anyP∈j1+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, whereTωΓΓ. 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/Γbecomes

0= {cγ+dδ∈C: 0≤c , d <1}. We define a linear mapL:Ω0

0by

L(u)=L(a+ib)=

x1 y1

x2 y2

a b

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We useLto define a measure-theoretic isomorphism between(Ωω,ω,µ,Tω)and

(Yω,Y,µY,Tω,L), whereYω=L(Ωω)⊂0,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,ω,µ,Tω)(Yω,Y,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∞,ω,ν,Rω)(Xd+,,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.

References

[1] J. Barnes,Application of Non-Invertible Ergadic Theory to Rational Maps of the Sphere, Ph.D. thesis, University of North Carolina, Chapel Hill, 1996.

[2] A. F. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. MR 92j:30026. Zbl 742.30002.

[3] L. E. Böettcher,The principal convergence laws for iterates and their applications to anal-ysis, Izv. Fiz.-Mat. Obshch pri Imper.13(1903), no. 1, 1–37.

[4] ,The principal convergence laws for iterates and their applications to analysis, Izv. Fiz.-Mat. Obshch pri Imper.14(1904), no. 3–4, 155–234.

[5] A. Freire, A. Lopes, and R. Mañé,An invariant measure for rational maps, Bol. Soc. Brasil. Mat.14(1983), no. 1, 45–62. MR 85m:58110b. Zbl 568.58027.

[6] L. Koss,Ergodic and Bernoulli Properties of Analytic Maps of Complex Projective Space, Ph.D. thesis, University of North Carolina, Chapel Hill, 1998.

[7] S. Lattès,Sur l’iteration des substitutions rationelles et les fonctions de Poincarè, C. R. Acad. Sci. Paris. Ser. I. Math166(1919), 26–28.

[8] M. Ju. Ljubich,Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynamical Systems 3 (1983), no. 3, 351–385. MR 85k:58049. Zbl 537.58035.

[9] R. Mañé,On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat.14(1983), no. 1, 27–43. MR 85m:58110a. Zbl 568.58028.

[10] ,On the Bernoulli property for rational maps, Ergodic Theory Dynamical Systems 5(1985), no. 1, 71–88. MR 86i:58082. Zbl 605.28011.

[11] , Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987. MR 88c:58040. Zbl 616.28007.

[12] W. Parry,Automorphisms of the Bernoulli endomorphism and a class of skew-products, Ergodic Theory Dynam. Systems 16 (1996), no. 3, 519–529. MR 97h:28006. Zbl 851.58030.

[13] K. Petersen, Ergodic Theory, Cambridge University Press, Cambridge, 1983. MR 87i:28002. Zbl 507.28010.

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[15] T. Ueda,In Topics Around Dynamical Systems, World Scientific Publ., 1993.

[16] A. Zdunik,Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math.99(1990), no. 3, 627–649. MR 90m:58120. Zbl 820.58038.

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

References

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