ONE-SIDED LEBESGUE BERNOULLI MAPS OF THE SPHERE OF DEGREE n 2 AND 2 n 2

We prove that there are families of rational maps of the sphere of degree n2 (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.


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 examples of such maps.A map is said to be one-sided Bernoulli of maximal entropy if it can be modeled by a fair d-sided die toss, where d is 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 every 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.
Let R ω , for ω = n or n + ni, denote the families of maps we consider where n ∈ N, and ωω ≥ 2. In Section 2, we describe how the maps R ω from these two families are constructed.In Section 3, we construct a toral map isomorphic to R ω which is used throughout the rest of this paper.In Section 4, the one-sided Lebesgue Bernoulli property is introduced and the method used to prove R ω is one-sided Bernoulli is presented.Then Sections 5 and 6 are devoted to proofs concerning the two families of maps, R n and R n+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.Let C ∞ denote the complex sphere and m denote the normalized surface area measure of C ∞ .Let α, β ∈ C − {0} be such that α/β is not a real number.We define a lattice of points in the complex plane by Γ = [α, β] = {αi + βj : i, j ∈ Z}.Given a lattice Γ and a complex number ω satisfying ωΓ ⊂ Γ , the map T ω (z) = ωz (mod Γ ) gives an endomorphism of the complex torus C/Γ .In this paper, we restrict to maps T ω , where Let ℘ Γ denote the Weierstrass elliptic function defined on a lattice Γ by (2.1) It is easy to see that ℘ Γ is an even, onto, 2-fold branched cover of C ∞ .Until Section 7 of this paper, we use Γ = [1,i], and we denote ℘ = ℘ [1,i] .We define a map R ω : The map R ω is well defined since T ω Γ ⊂ Γ , and hence the following diagram commutes: Since R ω is locally a composition of analytic maps, R ω is analytic.Further, all analytic maps of C ∞ are rational, so R ω is a rational map with degree d = ωω.Thus, for ω = n, we have the deg(R ω ) = n 2 and for ω It can be shown explicitly via algebraic identities that R 2 (z) = (z 2 +1) 2 /[4z(z 2 −1)] as done in [2].Also, Ueda [15] has proved that the rational map R 1+i of the sphere is given by the equation R 1+i (z) = (1/2i)(z − (1/z)), a degree 2 map.
We represent the torus C/Γ by a period parallelogram Ω 0 = {x + iy : x, y ∈ R, 0 ≤ x, y < 1}, the unit square without the top or right boundaries.Let Ꮾ 0 be the Lebesgue measurable sets in Ω 0 , and let leb be normalized 2-dimensional Lebesgue measure restricted to Ω 0 .We use the σ -algebra Ꮾ and measure ν on C ∞ induced by the factor map ℘.Thus Ꮾ is precisely the σ -algebra of Borel sets on C ∞ and the measure ν is given by ν(A) = leb(℘| −1 Ω 0 A) for all A ∈ Ꮾ.By definition, since T ω is measurepreserving with respect to leb, R ω is measure-preserving with respect to ν.We note that the measure ν is equivalent to m, the normalized surface area measure on C ∞ .
Since ℘ is an even map, ℘(u) = ℘(−u) for all u ∈ C. Therefore, the map R ω has critical values at precisely the points ℘(u) where u ≡ −u (mod Γ ), as R ω fails to have d distinct preimages at these points.There are exactly 4 points in the period parallelogram Ω 0 where u ≡ −u (mod Γ ).These are Therefore, the critical values (i.e., the images under R ω of the critical points) of the rational map R ω are The dynamical behavior of a rational map R ω depends on the postcritical set of R ω , which is defined as The postcritical orbit of the map R ω depends on whether n is even or odd.
For any ω = n or ω = n + ni with n even, If ω = n and n is odd, then In all cases, R ω has a finite postcritical set, the critical values are periodic or preperiodic, and none of the critical points of R ω are periodic (i.e., there are no critical points c of R ω and no integers r > 0 such that R r ω (c) = c).Therefore, R ω is ergodic and conservative (see [14]) as well as exact (see [1]) with respect to m.

Realization of R ω on the torus.
In order to study the behavior of R ω on the sphere, we analyze the behavior of an isomorphic map on a subset Ω ω of the topological closure of the period parallelogram Ω 0 as defined below.Due to the differences in the structure of P (R ω ) for n even and n odd, we define Ω ω separately for n even and n odd as follows: If n is odd, (3.1) See Figure 3.1 for an illustration of Ω ω .We next discuss some properties associated with Ω ω .For all u ∈ Ω 0 , we define an operation prime ( ) by u = 1 + i − u ∈ Ω 0 .Note that u = −u (mod Γ ) and u = u.Lemma 3.1.There is a set of leb-measure one in Ω 0 on which the following property holds: Proof.Let ∂Ω ω denote the boundary of Ω ω , and Ω 0 ω = Ω ω − ∂Ω ω denote the interior of Ω ω .Clearly, ∂Ω ω has leb-measure 0, so it suffices to prove this lemma for the points in Ω 0 ω .Suppose u ∈ Ω 0 ω .Then the midpoint of the line segment connecting u and The other direction works similarly.
Let θ ω = ℘| 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.Therefore, θ −1 ω (z) has a unique value for every and the following diagram commutes: Let Ꮾ ω be the Lebesgue measurable sets in Ω ω .We define µ(A) = 2 leb(A) for all A ∈ Ꮾ ω .Thus, T ω is well defined on a set of µ-measure 1 and T ω is measure preserving with respect to µ.Using the construction defined above, we are able to prove the following proposition which enables us to study the behavior of R ω by analyzing T ω on Ω ω .Proposition 3.2.We have Proof.Let U be a set of leb-measure 1 in Ω 0 given in Lemma 3.1, and define U ω = U ∩ Ω ω .Then θ ω is one-to-one on U ω .By the definitions of ν and µ, Combining these facts with the construction of T ω above, we have shown that θ ω : U ω → ℘(U ω ) is a measure-theoretic isomorphism, proving the proposition.

The one-sided Lebesgue Bernoulli property. Let
..,d−1} be the collection of one-sided sequences on d symbols.Let Ꮿ be the σ -algebra generated by cylinder sets of X + d and let σ be the one-sided shift on X + d given by σ (x 0 ,x 1 ,x 2 ,...)= (x 1 ,x 2 ,...).Let p d be the (1/d, 1/d,...,1/d) Bernoulli measure on X + d , i.e., p d ([j 0 ,j 1 ,..., We say that a map R ω is one-sided Bernoulli with respect to the measure ν if ). Due to Proposition 3.2, this is equivalent to satisfying (Ω ω , Ꮾ ω ,µ, T ω ) (X + d , Ꮿ,p d ,σ ).To prove that T ω is one-sided Bernoulli with respect to µ, we construct a partition ᏼ ω of the space Ω ω into d congruent atoms: A 0 ,A 1 ,...,A d−1 .Then we code the system using a map ϕ ω : The map ϕ ω is the standard method used to code a system from a partition.The join of two partitions η and ξ is defined to be η ∨ ξ = {A j ∩ B k : A j ∈ η, B k ∈ ξ}.We define ), and we use Proposition 4.1 to show that the map ϕ ω is an isomorphism.A proof of Proposition 4.1 can be found in Petersen [13]; see Barnes [1] for the noninvertible case.To show that ᏼ ω is a generating partition, i.e., ∞ j=0 ᏼ j ω is the Borel σ -algebra (mod 0), we apply Theorem 4.2, whose proof can be found in Mañé [11].Theorem 4.2.Let (X, Ᏸ,µ) be a probability space, where X is a locally complete separable metric space and Ᏸ is the Borel σ -algebra of X.Let ᏼ

The family R
We prove the following for R 2 .
Proof.In order to use the methods described in Section 4, we define a partition, ᏼ 2 of Ω 2 by ᏼ 2 = {A k : 0 ≤ k ≤ 3}, where (5.1) To determine the partitions ᏼ j 2 , we analyze images of the boundaries of the atoms of ᏼ 2 under T −1 2 .All of these boundaries are horizontal or vertical lines.Notice that any vertical line in Ω 2 can be expressed by l v,k = {u ∈ Ω 2 : Re(u) = k}, for some real number k. T −1 2 (l v,k ) is a pair of vertical lines described by {u ∈ Ω 2 : Re(u) = k/2 or Re(u) = (k + 1)/2}.Similarly, any horizontal line in Ω 2 is of the form l h,k = {u ∈ Ω 2 : Im(u) = k} for some k ∈ R and also has 2 lines in its image under T −1 2 .One of these lines is the horizontal line satisfying and for all u ∈ {z : Im(z) = 1 − k}, u ∈ {z : Re(z) = k}.In Figure 5.1, we illustrate the partitions ᏼ 2 and ᏼ 1 2 .
As j approaches ∞, the sequence of partitions consists of rectangles with smaller and smaller diameter.We can calculate Therefore, by Proposition 4.1 and Theorem 4.2, ϕ 2 is an isomorphism.
We extend the same type of argument to handle all n ∈ Z −{0, 1, −1}.
First we consider the case where n is 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, ᏼ n = {A k : 0 ≤ k ≤ n 2 − 1}, where (5. 3) The behavior of T n on ᏼ j n is similar to that of T 2 on ᏼ j 2 in that each atom of the partition ᏼ j n is a rectangle which decreases in diameter as j increases.For |n| > 2, the diameter of any atom of any partition ᏼ j n is strictly less than the diameter of an atom of ᏼ and ϕ n is an isomorphism when n is even.Now suppose n is odd.Then Ω n is a triangle instead of a rectangle, but it still has half the leb-measure of the period parallelogram Ω 0 .As in the even case, the key to the proof comes from defining the appropriate partition ᏼ n .
We define and (5.5) See Figure 5.2 for illustration of ᏼ 3 .This partition consists of right triangles which are contained in (1/n) × (1/n) squares.Furthermore, we can calculate and ϕ n is an isomorphism.
6.The family R n+ni .We next extend Theorem 5.2 to the family of rational maps of the sphere Proof.We begin by proving Theorem 6.1 in the case where n = 1 and then extend the result to the maps of higher degree.We define a partition ᏼ 1+i = {A 0 ,A 1 } of Ω 1+i as follows: Thus the action of T 1+i on a point u is to rotate it by π/4 and increase its magnitude by √ 2, and T −1 1+i rotates the point by −π/4 and shrinks its magnitude by √ 2. We show illustrations of ᏼ 1+i , ᏼ 1 1+i , and ᏼ 2 1+i in Figure 6.1.Notice that the composition T n • T 1+i = T n+ni gives a degree 2n 2 map of the torus.By using the above arguments and Theorem 5.2, we see that R n+ni is isomorphic to the one-sided (1/2n 2 ,...,1/2n 2 ) Bernoulli shift on 2n 2 symbols.We use L to define a measure-theoretic isomorphism between (Ω ω , Ꮾ ω ,µ,T ω ) and (Y ω , Ꮾ Y ,µ Y ,T ω,L ), where Y ω = L(Ω ω ) ⊂ Ω 0 , Ꮾ Y is the Borel sets on Y ω , µ Y is normalized Lebesgue measure in Y ω , and T ω,L = LT ω L −1 .We state this result in the following lemma.This technique is further developed by Koss [6] to analyze other classes of critically finite rational maps with parabolic orbifolds.