A SPECIAL CLASS OF INFINITE MEASURE-PRESERVING QUADRATIC RATIONAL MAPS RACHEL BAYLESS AND JANE HAWKINS

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1 A SPECIAL CLASS OF INFINITE MEASURE-PRESERVING QUADRATIC RATIONAL MAPS RACHEL BAYLESS AND JANE HAWKINS Abstract. We show the existence of a set of positive measure in a slice of parameter space of quadratic rational maps for which there exists an invariant infinite σ-finite measure equivalent to a conformal measure. These maps are conservative, ergodic and exact with respect to the invariant measure. The maps are perturbations of generalized Boole maps and inner functions. For examples with real coefficients we calculate the Krengel entropy and show that it provides a complete invariant for c-isomorphism classes of quadratic maps of this type. 1. Introduction In this paper we prove the existence of a set of positive measure in a slice of parameter space of quadratic rational maps for which there exists an invariant infinite σ-finite measure equivalent to a uniquely determined conformal measure. These maps are ergodic and exact with respect to this conformal measure, and are related to generalized Boole maps and classical inner functions. If the coefficients are real we calculate the Krengel entropy, and in the quadratic case we show that Krengel entropy provides a complete invariant for c-isomorphism classes of these maps. Dynamical systems that exhibit extremely chaotic behavior by virtue of being ergodic and exact with respect to an infinite invariant measure, are somewhat rare. However, certain real analytic maps of the real line that preserve Lebegue measure provide some natural examples, and the earliest study is due to Boole [8], with ergodicity being established over 100 years later by Adler and Weiss [4]. These examples extend to inner functions on C, and their properties have been studied by many, including [1], [3], [11], and [12]. Since Boole functions are rational maps, they can be viewed as maps of the Riemann sphere, which we will denote by C. p In this paper we study a class of analytic functions that extends our measure theoretic analysis of Boole functions beyond the real line to the complex plane. Negative generalized Boole functions, which we will call Boole functions for ease of notation, are defined by (1.1) Spzq z β with all t k distinct. Nÿ k 1 p k t k z, β, t k, p k P R, p k ą 0, Date: June 16, Mathematics Subject Classification. 37F10, 32A20, 30D05, FIX. Key words and phrases. complex dynamics, Julia sets, infinite measure. 1

2 2 RACHEL BAYLESS AND JANE HAWKINS Any Boole function of the form (1.1), is a square root of an inner function, that is, S 2 S S maps the upper half plane to itself and preserves the real line. While traditionally these functions are viewed as maps of the unit disk preserving the boundary, they also provide natural examples of transformations that preserve Lebesgue measure, λ, on R. The first author studied Boole functions of the form (1.1) in [5] and [6], and proved that they are ergodic and exact with respect to λ. Furthermore, a formula for their Krengel entropy, an entropy that is well-defined for infinite measure preserving maps, was established in [5]. In this paper, we first show that Boole functions of the form (1.1), when viewed as maps of p C, have Julia set R Y t8u which we denote by p R. We then restrict to the real line and investigate the relationship between the Krengel entropy and isomorphic maps in the context of the infinite measure λ. Since infinite measures can be scaled, we must consider a less restrictive type of isomorphism called a c-isomorphism (defined in Section 2, where c is a constant and refers to the measure cλ.) We observe that since quadratic Boole functions admit no periodic orbits of period 2, they are conformally conjugate to maps in the unique parameter space of rational maps with this property ([7], [9]). We further use this conformally conjugate form to provide a characterization of the c-isomorphism classes of quadratic Boole functions and pinpoint in what way Krengel entropy provides a complete invariant of c-isomorphism. In Section 3 we consider Boole functions where the parameters, β, t k and p k are complex. We call these maps complex Boole functions. When the parameters are close to the real axis, the Julia set changes from a circle ( p R), to a quasi-circle, and the infinite invariant measure changes from λ to an infinite invariant measure µ that is equivalent to an appropriate conformal measure. A complex Boole function is no longer a square root of an inner function, but when viewed as a rational map restricted to its Julia set, many of the dynamical properties carry over. In particular, we show that many complex Boole functions are conservative, exact and ergodic with respect to µ, and that there is an open set of parameters corresponding to maps with these properties contained in a well-studied slice of quadratic parameter space Background and notation. By px, B, µ, T q we denote a nonsingular measurable dynamical system; we always assume T : px, B, µq Ñ px, B, µq, where X is a topological space, B is the σ-algebra of Borel sets, and µ is a σ-finite measure on X. In the definitions that follow, all sets A are measurable and each statement holds up to sets of µ measure 0. We assume T maps X onto X and µpaq 0 if and only if µpt 1 Aq 0 (i.e., T is nonsingular). We say T is conservative if there exists some n P N such that µpt n A X Aq ą 0 for every set A with µpaq ą 0. The map T is ergodic if T 1 paq A implies µpaq 0 or µpxzaq 0. We say T is exact if X Ť ně1 T n T n paq for every A with µpaq ą 0. In [6] it was shown that pr, B, λ, Sq, with S any real Boole function of the form (1.1), is λ-preserving, conservative, ergodic, and exact. The map T is n-to-1 if for almost every x P X, the set T 1 pxq contains precisely n distinct points. Given a nonsingular n-to-1 transformation, we define P tp i u n i 1 to be a Rohlin partition of X if T : P i Ñ X is one-to-one and onto for each i 1,..., n and

3 INFINITE MEASURE 3 Ť n i 1 P i X mod µ. Given px, B, µ, T q with Rohlin partition P tp i u n i 1, we denote each branch T Pi by T i. The Jacobian of T is defined by J T pxq ř n i 1 1 P i pxq dµt i pxq, dµ where 1 A denotes the characteristic function of the set A. Clearly if X R, µ λ, and T is piecewise C 1, then J T pxq T 1 pxq. A set A P B is called a sweep-out set for T if Ť 8 n 0 T n A X mod µ. For x P A we let φ A pxq be the first-return-time of x to A, so φ A pxq mintn : T n pxq P Au. The induced transformation, T A : A Ñ A, is defined by T A pxq T φ Apxq pxq for x P A. If px, B, µ, T q is a measure-preserving system and A is a sweep-out set for T, then T A is a measure-preserving transformation of pa, B A, µ A q, where B A tb XA : B P Bu and µ A pbq µpa X Bq. In [10] Krengel gave the following definition of entropy for infinite measure-preserving transformations. Let px, B, µ, T q be a conservative σ-finite measure-preserving system. If A P B is such that 0 ă µpaq ă 8, and A is a sweep-out set for T, then h Kr pt q hpt A, µ A q (i.e. the traditional Kolmogorov-Sinai entropy of the induced system). Letac proved that if G : pr, B, λq Ñ pr, B, λq is a rational function that preserves λ on R, then G is of the form G S for some S of the form (1.1) [11]. A formula for the Krengel entropy of pr, B, λ, Sq, where S is any Boole function of the form (1.1), was given in [6]. In fact, a more general result was shown. Theorem 1.1 ([6]). Any G : pr, B, λq Ñ pr, B, λq which is conservative, rational, and preserves λ on R has Krengel entropy given by: ż (1.2) h Kr pgq log G 1 pxq dλpxq. R We now change our focus from restricting the domain of Boole functions to R, and allow the domain to be the entire Riemann sphere p C. We present a basic result about the Julia sets for Boole functions. Definition 1.2. Let R be rational function of degree d ě 2, defined on p C. The Fatou set of R, F prq, is the open subset of p C defined by: tz P p C : tr n u is equicontinuous at zu. The Julia set of R, JpRq, is the complement of F prq in p C. When the obvious extension of a Boole function S to a map of the Riemann sphere is considered, we have the following result. By p R we denote the great circle on p C whose image is the real line under stereographic projection. Lemma 1.3 ([5]). For each S of the form (1.1), every point in R has precisely degpsq distinct preimages in R. Proof. We relabel the set of poles tt 1, t 2,..., t N u to be in increasing order so that s i ă s i`1 for i 1,..., N 1, and s i t k for some k. Then it is straightforward to check that P tp8, s 1 q, ps 1, s 2 q,..., ps N 1, s N q, ps N, 8qu is a Rohlin partition for S restricted to R and S maps each interval in P diffeomorphically onto R. There are N ` 1 intervals in P, and this is exactly degpsq.

4 4 RACHEL BAYLESS AND JANE HAWKINS Figure 1. A typical graph of a negative Boole transformation restricted to R Lemma 1.3 is illustrated in Figure 1. Proposition 1.4. Every Boole function of the form (1.1) satisfies JpRq p R. Proof. We first remark that each S of the form (1.1) viewed on C, p maps R onto R p and the point at 8 is a fixed point with multiplier 1. Moreover by Lemma 1.3, each x P R has degpsq preimages in R. Therefore R p is closed and completely invariant under S, so JpSq Ă R. p We now suppose there exists an interval I Ă F psq X R and fix some x 0 P I. Then given any 0 ă ε ă 1, there exists a δ ą 0 such that x x 0 ă δ ñ S n pxq S n px 0 q ă ε for all n P N. Consider the interval U rx 0, x 0 ` δ{2s; then U Ă I, and λpuq δ{2. Also U cannot contain a pole or a prepole of S since 8 P JpSq. By invariance of λ, λps n Aq λpaq for all Borel sets A Ă R, and by exactness, ď (1.3) ps n ps n puqqq R npn up to a set of λ measure 0. We have that λps n Uq λps n ps n puqqq, and S n puq is an interval in R. Therefore by (1.3) there exists some N such that λps N puqq λps N S N puq ą 1; by piecewise monotonicity of S, S N puq ps N px 0 q, S N px 0 ` δ{2qq or ps N px 0 ` δ{2q, S N px 0 qq. This contradicts equicontinuity, so there is no interval U Ă pf psqxrq. Clearly the point at 8 is fixed under any positive or negative generalized Boole function, and for maps of the form (1.1) we easily compute that the fixed point at 8 has multiplier 1. There are N other preimages of 8, namely the set of poles, tt 1,..., t N u. We refer the reader to a standard reference like [7] or [16] for background on parabolic fixed points for rational maps. We can draw the following conclusion from Proposition 1.4; (we prove a more general result of this form in Thm 3.2). Corollary 1.5. If S is a rational map of the form (1.1), then the parabolic fixed point at 8 has two petals.

5 INFINITE MEASURE 5 Proof. There are at least two petals since by exactness of S there are two repelling directions along R, one on each side of the point at 8. If there were more petals, then there would be more repelling directions, and JpSq would have to contain more points than just R. 2. Isomorphisms of negative Boole functions In traditional ergodic theory, entropy is an isomorphism invariant for probabilitypreserving transformations, but in the infinite setting we must account for a less restrictive type of isomorphism called a c-isomorphism. Definition 2.1. Let px 1, B 1, m 1, T 1 q and px 2, B 2, m 2, T 2 q be two infinite-measurepreserving systems. Suppose we have two sets M 1 P B 1 and M 2 P B 2 with m 1 px 1 zm 1 q 0 and m 2 px 2 zm 2 q 0 such that T 1 pm 1 q Ď M 1 and T 2 pm 2 q Ď M 2. For c P p0, 8s we say px 1, B 1, m 1, T 1 q is c-isomorphic to px 2, B 2, m 2, T 2 q if there exists an invertible map φ : M 1 Ñ M 2 such that for all A P B 2 M2, (1) φ 1 paq P B 1 M1, (2) m 1 pφ 1 paqq c m 2 paq, and (3) pφ T 1 qpxq pt 2 φqpxq for all x P M 1. We will denote this situation by φ : T 1 Ñ c T 2, and call φ a c-isomorphism. In this section, we consider maps of the form (1.1), and we restrict the domain to R. We have the following relationship between c-isomorphisms and Krengel entropy for Boole maps on pr, λq. Proposition 2.2. If S 1 and S 2 are two Boole transformations of the form (1.1), and φ : S 1 Ñ c S 2 is a c-isomorphism, then (2.1) h Kr ps 1 q c h Kr ps 2 q. Proof. Suppose φ : S 1 Ñ c S 2 is a c-isomorphism. By definition, S 2 φ S 1 φ 1 and λ φ 1 c λ. By (1.2) and the fact that pφ 1 q 1 pxq 1{pφ 1pφ 1 pxqqq we have, ż c h Kr ps 2 q c log S2pxq dλpxq 1 R ż c `log φ 1 ps pφ 1 1 xqq ` log S 1pφ 1 1 xq log φ xq 1 pφ 1 dλpxq. R By applying the substitution u φ 1 pxq and recalling that S 1 preserves λ, the last line becomes h Kr ps 1 q. Corollary 2.3. Krengel entropy is an isomorphism invariant for Boole transformations. That is, if S 1 and S 2 are 1-isomorphic, then h Kr ps 1 q h Kr ps 2 q. Corollary 2.4. If S 1 and S 2 are two Boole transformations, then there is at most one c P p0, 8s such that φ : S 1 Ñ c S 2 is a c-isomorphism. Proof. If S 1 is c-isomorphic to S 2, then h Kr ps 1 q c h Kr ps 2 q. The Krengel entropy of S 1 with respect to λ is a fixed value (similarly for S 2 ). Thus, the c must be unique.

6 6 RACHEL BAYLESS AND JANE HAWKINS Thus, if h Kr ps 1 q h Kr ps 2 q, then S 1 and S 2 are not 1-isomorphic, but they may still be c-isomorphic for some c 1. In the event that they are c-isomorphic, then the c is unique Quadratic case. In this section, we focus on quadratic Boole transformations, which we write as: (2.2) S pβ,p,tq pxq x β p t x, where β, p, t P R and p ą 0. Lemma 2.5. The Krengel entropy of S pβ,p,tq is 2π? p. Proof. By Theorem 1.1 we have ż h Kr ps pβ,p,tq q R ˆ p log 1 ` dλpxq. pt xq 2 Integration by parts followed by a substitution yields ˆ ˆt x 2p arctan? ` log 1 ` p and evaluating the above limits yields the result. p pt xq 2 j 8 px tq The next result shows that there is some symmetry with regard to the placement of the poles and constant. Lemma 2.6. The Boole function, S pβ,p,tq, is 1-isomorphic to S p β,p, tq. Proof. If we set ηpxq x, then η η 1 and η S pβ,p,tq η S p β,p, tq. We say a quadratic Boole transformation is normalized if it has the form S pβ,1,0q, with β P R. Theorem 2.7. Every quadratic Boole function, S pβ,p,tq, is? p-isomorphic to a unique normalized map S p ˆβ,1,0q, where ˆβ ˇ? p ˇˇˇ ě 0. ˇ 2t`β Proof. We first move the pole from t to 0 via a 1-isomorphism with conjugating map (2.3) ψ t pxq x t and ψ 1 t pxq x ` t. That is, pψ t S pβ,p,tq ψ 1 t q S p2t`β,p,0q. We now change the multiplier from p to 1 via a? p-isomorphism with conjugating map (2.4) ζ p pxq x? p and ζ 1 p pxq? p x. That is, ζ p S p2t`β,p,0q ζ 1 p S p 2t`β? p,1,0q. Lemma 2.6 implies that S pβ,1,0q, is 1-isomorphic to S p β,1,0q, so we choose ˆβ 2t`β? p to be nonnegative. 8,

7 INFINITE MEASURE 7 Corollary 2.8. Every quadratic Boole function, S pβ,p,tq, is? p-isomorphic to a Boole function T with h Kr pt q 2π. Proof. Consider the normalized form: T S p ˆβ,1,0q with ˆβ 2t`β? p from Theorem 2.7, which satisfies h Kr pt q 2π. Corollary 2.8 shows that Krengel entropy appears to be unable to distinguish among maps of the form (2.2) if all 3 parameters are allowed to move. However, in order to link this study to parametrized families of rational maps, we show that the values of ˆβ distinguish c-isomorphism classes, which is equivalent to showing Krengel entropy can distinguish c-isomorphism classes of quadratic negative Boole functions. Proposition 2.9. Let S pβ,1,0q and S pγ,1,0q be two quadratic Boole transformations. Then S pβ,1,0q is c-isomorphic to S pγ,1,0q if and only if β γ. In this case, c 1, and the conjugating map φpxq x λ-a.e., or φpxq x λ-a.e. Proof. (ùñ) If φ : S pβ,1,0q Ñ c S pγ,1,0q is a c-isomorphism, then pφ S pβ,1,0q qpxq ps pγ,1,0q φqpxq, for almost every x P R. If we consider the Jacobian (the local Radon-Nikodym derivative) of both sides at x, and note that dλφ pxq 1{c, for λ-a.e. x, then we have dλ Spβ,1,0q 1 pxq S1 pγ,1,0qpφpxqq, λ-a.e. Therefore, 1 ` 1 x 1 ` 1, for λ-a.e. x P R, 2 pφpxqq2 so pφpxqq 2 x 2, and therefore φpxq x λ-a.e. Let M Ď R be a measurable set such that φ S pβ,1,0q S pγ,1,0q φ on M and λprzmq 0. Define A tx P M : φpxq xu and B tx P M : φpxq xu. Note that λprzpa Y Bqq 0. Without loss of generality assume λpaq ą 0. If y P S 1 pβ,1,0qpbq X A, then writing out each side of the equation pφ S pβ,1,0q qpyq ps pγ,1,0q φqpyq, we have (2.5) y ` β 1 y y γ ` 1 y. Simplifying yields 2y 2 `pβ `γqy 2 0, and there are at most two points y for which (2.5) holds. Thus, there are at most two points in S 1 pβ,1,0qpbq X A, so S 1 pβ,1,0q pbq Ă B λ-a.e. A similar calculation shows that B Ă S 1 pβ,1,0qpbq, so B is invariant modulo sets of λ measure 0. By ergodicity of S pβ,1,0q, either λpbq 0 or λprzbq 0. By assumption the first case holds; therefore, β γ. If we assume instead that λpbq ą 0, then a similar argument proves that β γ. (ðù) This direction is obvious using Lemma 2.6.

8 8 RACHEL BAYLESS AND JANE HAWKINS Corollary The representation of S pβ,p,tq in the form S pβ 1,1,0q, where β 1 ą 0 is unique. Proof. Assume φ 1 : S pβ,p,tq Ñ c1 S pβ 1,1,0q is a c 1 -isomorphism and φ : S Ñ c pβ,p,tq S p β,1,0q is a c-isomorphism, where both β 1 ą 0 and β ą 0. Then φ 1 φ : S p β,1,0q Ñ S pβ 1,1,0q is a c 1 -isomorphism, and by Proposition 2.9 c c1 c and β 1 β ˇ 2t`β? p ˇˇˇ. Remark (1) Each S pβ,p,tq has a fixed point at 8 with multiplier 1, so by Milnor [15], S pβ,p,tq lacks period two orbits (apart from the fixed points). (2) Viewing S as a map on the sphere, we know by [9] that S is conformally conjugate to a map of the form R a z2 z, for a P Czt 1u. 1`az (3) If a ă 1 then JpR a q R, p and if a ă 3, then R a is conformally conjugate to an R a 1 with a 1 P r 3, 1q, (see [7],[9]). (4) Using the spherical metric on C p and the isometric involution ψpzq 1, for z each a P r 3, 1q, ψ R a ψ 1 H b pzq S p b,b,1q pzq z ` b b 1 z, where b pa ` 1q P p0, 2s. (5) If b b 1, then h Kr ph b q h Kr ph b 1q. 2 b (6) The normalization of H b is S p 2 b?,1,0q. As b decreases from 2 to 0,? b b increases monotonically from 0 to `8. Conversely, if β ě 0, S pβ,1,0q is? 1 b -isomorphic to H b, where b 1 β 2 ` 4 β a β 2 2 ` 8 P p0, 2s. When b b 1 P p0, 2s, we already know that H b and H b 1 are not conformally conjugate rational maps of C. p We now show that each H b corresponds to a unique c-isomorphism class of quadratic negative Boole functions. Theorem Each map H b S p b,b,1q pzq z ` b b, with b P p0, 2s, satisfies 1 z the following: (1) h Kr ph b q 2π? b; (2) every quadratic negative Boole function is c-isomorphic to a function H b ; (3) S pβ1,p 1,t 1 q is c-isomorphic to S pβ2,p 2,t 2 q if and only if they are both c-isomorphic to the map same map H b. Proof. (1) follows immediately from Lemma 2.5. (2) follows using Theorem 2.7 and Remark 2.11 part (6). (3) (ñ) By Theorem 2.7, for j 1, 2, S pβj,p j,t j q is? p j - isomorphic to S p ˆβj,1,0q with ˆβ 2tj`β j j? pj, chosen so that ˆβ j ě 0. Then by Proposition 2.9, ˆβ1 ˆβ 2, and (3) follows using Remark 2.11 part (6). (ð) If S pβ1,p 1,t 1 q and S pβ2,p 2,t 2 q are each c j -isomorphic to H b, then the composition of one isomorphism with the inverse of the other gives the c-isomorphism between them. Corollary Krengel entropy of H b is a complete invariant for c-isomorphism classes of quadratic negative Boole functions.

9 INFINITE MEASURE 9 We conclude this section by showing a complete c-isomorphism invariant as a set of values in the general setting (without normalization). Theorem Two quadratic Boole maps S pβ,p,tq and S pγ,q,sq are c-isomorphic if and only if (2.6) c h KrpS pβ,p,tq q and 2t ` β h Kr ps pγ,q,sq q ˇ? p ˇˇˇˇ 2s ` γ ˇ? q ˇˇˇˇ. Proof. (ùñ) Assume there exists a c-isomorphism, ξ : S pβ,p,tq Ñ c S pγ,q,sq. By Proposition 2.2 we have that h Kr ps pβ,p,tq q c h Kr ps pγ,q,sq q, so c h KrpS pβ,p,tq q h Kr ps pγ,q,sq. To show q ˇ? p ˇˇˇ 2s`γ ˇˇˇ? q ˇˇˇ we transform Spβ,p,tq and S pγ,q,sq to their completely normalized forms. ˇ 2t`β As in Theorem 2.7, we have (2.7) ζ p ψ t : S pβ,p,tq Ñ?p S p ˆβ,1,0q and ζ q ψ s : S pγ,q,sq Ñ?q S pˆγ,1,0q, where ˆβ ˇ 2t`β? p ˇˇˇ and ˆγ 2s`γ ˇˇˇ? q ˇˇˇ. Thus, there exists a map φ ζq ψ s ξ ψ 1 t ζ 1 p : S p ˆβ,1,0q Ñĉ S pˆγ,1,0q, and by Proposition 2.9 ĉ 1 and ˆβ ˆγ. (ðù) Assume that ˇ 2t`β? p ˇˇˇ 2s`γ ˇˇˇ? q ˇˇˇ, and transform Spβ,p,tq and S pγ,q,sq to their completely normalized forms. By Proposition 2.9 there exists a 1-isomorphism φ : S p ˆβ,1,0q Ñ 1 S pˆγ,1,0q, because ˆβ ˆγ. Therefore, ξ ψ 1 s ζ 1 q φ ζ p ψ t : S pβ,p,tq Ñ c S pγ,q,sq and c? p? q h KrpS pβ,p,tq q. h Kr ps pγ,q,sq q 3. Negative Boole functions with complex coefficients In this section we consider degree d maps of the form: (3.1) Spzq z γ d 1 ÿ k 1 ρ k τ k z, γ, τ k, ρ k P C, Repρ k q ą 0. We will call these complex (negative) Boole functions. When the imaginary parts of the coefficients above are not too large, complex Boole functions retain many of the dynamical and measure theoretic properties of real Boole functions, using an appropriately modified measure. The main result of this section is Theorem 3.8; we show that there is an open set of parameters in parameter space that correspond to quadratic complex Boole functions with an infinite invariant measure equivalent to conformal measure. Let R : Ĉ Ñ Ĉ be a rational map of degree d ě 2. We say R is parabolic if JpRq contains rationally indifferent periodic points but contains no critical points. All Boole functions of the form (1.1) are parabolic; in this section we study complex perturbations of these maps that remain parabolic. It is possible to move the coefficients enough so that a critical point lands on a prepole; for example, the map H 2i, using the notation of Theorem 2.12 is not parabolic since the critical point c i lands on the fixed point at 8.

10 10 RACHEL BAYLESS AND JANE HAWKINS If we consider a complex Boole function S of the form (3.1) where all the coefficients are sufficiently close to the real axis, then JpSq will no longer be a circle, but a homeomorphic (quasiconformal) image of R 8, called a quasi-circle (see Theorem 3.6 below and the general theory in [13]). The fixed point at 8 is a simple parabolic fixed point of S and has multiplicity 2p ` 1 pp ą 0q as a fixed point of S 2, with S 2 having multiplier 1 at 8. Then 8 as a fixed point of S has p distinct immediate basins of attraction, each of which consists of 2 disjoint Fatou components forming a period 2 cycle. We call each component of the immediate basin of attraction a Leau domain L 1, L 2,..., L 2p (see, eg. [16]). As for the real coefficient case, the attracting basin of 8 is the open set of points in p C converging to the fixed point. It is straightforward to calculate that the attracting directions for 8 are along the imaginary axis. The rays of symmetry of the attracting petals of a rational map of the form (3.1) near 8 are given by the next two theorems, (see, e.g., [16], Lemma 10.1 and its proof) Theorem 3.1. [16] Suppose Rpzq zp1 ` αz p ` phigher order termsqq, for some α 0, p P N, near the origin. Then there exist p attracting petals, or equivalently p evenly spaced attracting directions at the origin and any point that approaches the rationally neutral fixed point at 0 must approach it in one of these p directions. The vectors v j, j 1,..., p, that give the attracting directions are determined by the 1{p ˆ 1 solutions v to the equation: pαv p 1; i.e., each v j satisfies v j. pα There are also p repelling petals which are defined to be the attracting petals for R 1 near a neutral fixed point, and the repelling directions are obtained by finding w such that pαw p 1. The following result shows that Theorem 3.1 implies there are two petals for the Boole functions considered in this paper. Theorem 3.2. If S of the form (3.1), then the fixed point at 8 has two petals, with attracting directions determined by the two vectors satisfying v p 1{p4 ř d 1 k 1 ρ kqq 1{2. Proof. For the first part of the result we want to show that 8 is a fixed point of S 2 of multiplicity 3. If we set gpzq S Spzq, and then consider Gpzq 1{gp 1 q, then z the theorem is equivalent to showing that near 0, (3.2) Gpzq zp1 c 1 z 2 c 2 z 3 q, with c 1 2 ř d 1 1 ρ k 0. Then using α c 1 and p 2, we apply Thm 3.1 to obtain the second statement of the theorem. We claim Eqn (3.2) is equivalent, for z ă 1, to (3.3) Gpzq z 1 ` pc 1 z 2 ` c 2 z 3 ` c 3 z 4 q.

11 INFINITE MEASURE 11 1 This is since for w ă 1, 1 ` w 1 w ` w2 w 3 `. Inverting the expression in Eqn (3.3), it is equivalent to show for all 0 ă z ă 1, ˆ1 (3.4) g 1 z z p1 ` c 1z 2 ` c 2 z 3 ` q 1 z ` c 1z ` c 2 z 2 `. Therefore to show Eqn(3.2), it is enough to show that for z ě R, i.e., near 8, (3.5) gpzq z ` c1 z ` c2 z 2 `, and to compute c 1. Eqn (3.5) is just the Laurent series expansion of S 2 about the fixed point at 8. We use this simple fact about series: for p ą 0, τ P C, and z ą R, R large, 8ÿ (3.6) p τ z p z n 0 τ z n p z p1 ` τ z ` q Using Eqn(3.6) we start with the Laurent series centered at 8 of S; it is not hard to show there is an R ą 0 such that for z ą R: (3.7) Spzq z γ ` ř k ρ k z d 1 ÿ ` ř k ρ kτ k z 2 ` ř k ρ kτ 2 k z 3 ` d 1 ρ k By definition we have: gpzq z ` τ k 1 k z ÿ ρ k ; and for z large, τ k 1 k Spzq ˆ1 Spzq z γ ` O. z Defining δ k pτ k ` γ ` ɛq, for small ɛ ɛprq, and applying Eqn(3.6) (3.8) gpzq z p ř k ρ k z ` ř k ρ kτ k z 2 ` ř k ρ kτk 2 d 1 ÿ ` q z 3 k 1 ρ k τ k Spzq, which, after applying (3.6) once more to the right most term, gives ř k (3.9) gpzq z p ρ ř k k ` ρ ř k k q p ρ ř kτ k k ρ kδ k q ` ; z z z 2 z 2 this establishes Eqn(3.5) with c 1 2 ř d 1 1 ρ k 0. Parabolic rational maps admit natural measures which play the role of λ on JpSq, called conformal measures. Let R be a rational map of degree ě 2; for h ą 0 a probability measure m h on JpRq is called h-conformal for R if ż (3.10) m h prpaqq R 1 h dm h for every Borel set A Ă JpRq such that R A is injective. The following result is a combination of several results from [2] (including Theorems 3.1, 8.7, and 8.8). A

12 12 RACHEL BAYLESS AND JANE HAWKINS Theorem 3.3. Assume that R is a parabolic rational map. Then there exists a unique h-conformal measure m h for R with the following properties: (1) m h is nonatomic; (2) there exists an R-invariant measure µ m h that is σ-finite and ergodic; (3) h HDpJpRqq ă 2, where HDpJpRqq denotes the Hausdorff dimension of JpRq. For ω P JpRq a rationally indifferent periodic point of period k with ppωq petals, let ppωq ` 1 (3.11) α min h. ω ppωq Theorem 3.4 ([2], Theorems 9.8 and 9.11). For a parabolic map R, there exists a conservative and exact σ-finite infinite invariant measure, µ m h, if and only if α ď 2. Theorem 3.5. Assume S is a complex Boole function (of the form (3.1)), where the fixed point at 8 is the only parabolic cycle. If HDpJpSqq ă 4, then there exists a conservative and exact σ-finite infinite invariant measure, µ m h, with h 3 HDpJpRqq. Proof. By Theorem 3.2 there are two petals at 8, so we apply Eqn (3.11) and Theorem 3.4 to obtain the result. It remains to show that many complex Boole functions satisfy the hypotheses of Corollary 3.5. Since for Boole functions of the form (1.1), HDpJpSqq 1, it suffices to establish some continuity properties Sequences converging to real Boole functions. We fix a real Boole function of the form (1.1) with parameters pβ, t 1,..., t N, p 1,..., p N q Ă R 2N`1 and p k ą 0. We choose any sequence ts n u of complex Boole functions, each of degree N `1, satisfying for each n P N: Nÿ S n z γ pnq ρ pnq k k 1 τ pnq k z, γpnq, τ pnq k, ρ pnq k j k; S n Ñ S in the sense that P C, Repρ pnq pnq q ą 0, τ k pγ pnq, τ pnq 1,..., t pnq N, ρpnq 1,..., ρ pnq N q Ñ pβ, t 1,..., t N, p 1,..., p N q j τ pnq k (S n converges to S algebraically [14]); S n has no critical points in JpS n q and no other parabolic points apart from the one at z 0 8. Theorem 3.6 ([14], Theorems 1.3 and 1.4). Under the assumptions above, JpS n q Ñ JpSq in the Hausdorff topology, and HDpJpS n qq Ñ 1 HDpJpSqq. In particular for n large enough, HDpJpS n qq ă 4{3. if

13 INFINITE MEASURE The quadratic case. We now turn to quadratic complex Boole functions of the form (3.12) Spzq z γ ρ, γ, τ, ρ P C, Repρq ą 0. τ z Theorem 3.2 shows there are two petals at 8. We prove that on a large set of parameters the measure µ from Theorem 3.3 is infinite and has highly mixing measure theoretic properties (eg., conservative, exact, and ergodic), sharing the properties of Boole functions. In Section 2 (Remark 2.11 and Theorem 2.12) we showed that every quadratic negative Boole function on R is c-isomorphic to precisely one map of the form S p b,b,1q, where b P p0, 2s, and each S p b,b,1q is conformally conjugate to exactly one map R a pzq z 2 z, where a P r 3, 1q, Moreover if we define Ω Czt 1u, for a P Ω, it is easy to 1`az see that R a is conformally conjugate to R a 1 with a 1 3 a Proposition 3.7. Given any a P r 3, 1q, there exists a neighborhood V a Ă Ω of a such that every η P V a corresponds to a Boole function of the form (3.12). Proof. For any η P V a, set b p1`ηq; we conjugate R η to H b pzq z`b b 1 z using ψpzq 1 z. It is easy to see that we can choose V a such that Repbq Rep p1`ηqq ą 0. Theorem 3.8. There exists an open set U Ă Ω such that for each a P U, the complex Boole map H p1`aq pzq z p1 ` aq p1`aq admits a nonatomic conservative 1 z and exact σ-finite infinite measure equivalent to the conformal measure m h, with h HDpH p1`aq q. Proof. We consider a P Ω, and set b p1`aq; we use Proposition 3.7 to parametrize Boole functions H b with this parameter. By Theorem 3.6 there exist open sets U a Ă Ω around each a ă 1 where the Hausdorff dimension of H b varies continuously with the parameter, and HDpH b q ă 4{3 for all a P U a. Therefore, by Theorem 3.4 we have the result for all a P U a X V a, and we set U ď pu a X V a q. apr 3, 1q 1`a. We illustrate the parameter space for quadratic Boole functions in Figure 2. References 1. Aaronson, J (1978) Ergodic theory for inner functions of the upper half plane. Ann. Inst. H. Poincaré Sect. B (N.S.) 14, no. 3, Aaronson J, Denker M, Urbanski M (1993) Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337, no. 2, Aaronson J, Park, K (2009) Predictability, entropy and information of infinite transformations. Fund. Math. Vol. 206, no. 2, Adler, R, Weiss B (1973) The ergodic infinite measure preserving transformation of Boole, Israel J. Math, Bayless R (2013) Entropy of Infinite Measure-Preserving Transformations, PhD thesis, University of North Carolina at Chapel Hill.

14 14 RACHEL BAYLESS AND JANE HAWKINS Figure 2. In a-space for the family of maps R a or equivalently H p1`aq, the blue line represents the real Boole functions and the light shaded region shows where parameters correspond to maps with infinite invariant measures. The parameters colored black show maps with an attracting fixed point. The dotted red circle encloses the region containing precisely one conformal equivalence class of each map. 6. Bayless R Ergodic Properties of Rational Functions that Preserve Lebesgue Measure, (preprint, 2016). 7. Beardon A (1991) Iterations of Rational Functions, Springer-Verlag GTM Boole G (1857) On the comparison of transcendence with certain applications to the theory of definite integrals. Philos. Trans. Roy. Soc. London. Vol. 147 Part III, Hagihara R (2009) Quadratic rational maps lacking period 2 orbits. Proc. Amer. Math. Soc. 137, no. 9, Krengel, U (1967) Entropy of conservative transformations. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Vol.7, Letac G (1977), Which functions preserve Cauchy laws? Proc. Amer. Math. Soc., Vol. 67, No. 2, Li, T Schweiger, F (1978), The generalized Boole s transformation is ergodic, Manuscripta Math., Vol. 25, No. 2, McMullen, C (1994) Complex dynamics and renormalization. Annals of Mathematics Studies, 135. Princeton University Press, Princeton, NJ. 14. McMullen, C (2000) Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps. Comment. Math. Helv. 75, no. 4, Milnor J (1993) Geometry and dynamics of quadratic rational maps, Experimental Math 2, Milnor J (2000) Dynamics in One Complex Variable: Friedr. Vieweg & Sohn, Verlagsgesellschaft mbh.

15 INFINITE MEASURE 15 Department of Mathematics, Agnes Scott College, 141 E. College Ave. Box 1093 Decatur, Georgia address: Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, North Carolina address: