zeta functions and asymptotic formulae for preperiodic orbits of hyperbolic rational maps

Math. Nachr. 186 (1997), 259-284

Zeta Functions and Asymptotic Formulae for Preperiodic Orbits of Hyperbolic Rational Maps

By SIMON WADDINGTON of Ghttingen

(Received February 20, 1995) (Revised Version March 7, 1997)

Abstract. For a hyperbolic rational map R of the Riemann sphere of degree d 2 2, restricted to its . l d b set J(R), we define a %eta function C R ( d ) , which counts the prepenodic orbib of R, according to Lhe weight function IR’I : J(R) -+ C . An analysis of the analytic domain of ( ~ ( d ) , using techniques from symbolic dynamics, yields weighted asymptotic formulae for the preperiodic orbits of R. We describe m application to diophantine number theory.

0. Introduction

The purpose of this paper is to study a new class of zeta functions for preperiodic orbits of expanding rational maps, and to derive asymptotic weighted counting formulae.

Let R be an expanding rational map of the Riemann sphere of degree d 2 2, and consider the restriction of R to the Julia set J(R) (see Section 2 ) . A point r E J ( R ) is called preperiodic under R if for some M ( z ) 2 1, each of the points I , Rr, . . . , R’(’)-’z is not periodic, but the point R’(’)z is periodic. Let P ( z ) be the least period of khe point RM(’)z, that is P ( r ) = k, where k is the least natural number such that R‘ (RM(”)z) = R‘(’)r, and define N ( r ) = M ( z ) + P ( z ) , which is precisely the number of points in the forward orbit of r . Preperiodic behaviour is of particular interest in t,he study of rational maps (see for example page 100 of [l], or [14]). (Preperiodic points are also refered to as eventually periodic points by some authors.)

We define a Dirichlet series, called the zeta function for R and denoted by ( ~ ( s ) . This function counts preperiodic points, according to the weight function IR’I : J(R) + IR

1991 Mathemafica Subject Classification. Primary 30D05, Secondary 58320. Keywords and phrases. btional maps of the Riemann sphere, dynamical zeta functions.

260

and is defined by

Math. Nachr. 186 (1997)

m

wherever the infinite summation converges. Let Fix(Rt) denote the set of fixed pointti of Rk. A point is in the finite set E(n) if there exists m with 1 5 rn 5 n - 1 such that Rjy is not periodic for 0 5 j 5 m-1, but Rmy E Fix(Rp) where p = n-m. This is an extension of the definition of the zeta function for periodic orbits of expanding maps, given by RUELLE, [18], [19]. The justification for calling the function ( ~ ( 8 ) a zeta function is that it plays a similar role to the usual dynamical zeta function in proving asymptotic orbit counting results. As the referee pointed out, there is no reasonablt, analogue of the Euler product representation for our zeta function. The problem of defining a zeta function for preperiodic points was raised by S. M. HEINEMANN, and was communicated to me by M. DENKER.

An important property of expanding rational maps is the existence of finite Markov. partitions. This allows us to study zeta functions at the level of symbolic dynamics. Our main tool is an extension theorem for a very general zeta function which encodes the preperiodic orbits of the shift map. This result is analogous to a result of PARRY and POLLICOTT, (161 Theorem 5.6, for the zeta function for periodic orbits. We USP

this theorem to study the symbolic dynamic analogue (,(s) of (~(5) .

Weshow that (,,(s) is analyticin a halfplane {s : &(s) > 6 } , where 6 = dimH(J(R)) is the Hausdorff dimension of J ( R ) . In order to determine the behaviour of (,,(s) 011

the critical line {s : Re(s) = 6 } , we partition all rational maps into two clasees, labelled (I) and (II), respectively. This is analogous to the condition that an Anosov flow is weak mixing or not weak mixing, respectively. We ehow that almost all maps are of type (I), with respect to the Lebesgue measure on the parameter space of coeflicients of R (see Proposition 2). For maps of type (I), c,,(s) has a pole of order 2 at s = 6, and is otherwise analytic in a neighbourhood of the critical line. By applying a Tauberian Theorem due to DELANGE [4], we deduce the following theorem.

Theorem 0.1. I f R is ofiype (I) inen

# { z E J (R) : z eueniually periodic, l(RN(z))'(z)l 5 t } - Ct6 logi as t - 00, where 6 = dimH(J(R)) and C > 0 as an explicit consiani.

This result occurs as Theorem 6.5 in the text, in a slightly different notation. A similar result also holds for maps of type (11). Since we only know a few relatively simple examples of maps of type (11), we do not discuss this case in detail.

We also remark that all the results concerning the analytic domain of c d ( s ) carry over to ( ~ ( 8 ) (see Theorem 7.1).

We have chosen to study zeta functions in the prototypical example of hyperbolic rational maps. It would also be interesting, although more difficult, to study the function ( ~ ( s ) in the case that there are critical points of R in the Julia set. Some related results and conjectures for the Ruelle zeta function have been given by HINKKANEN

Waddington, Preperiodic Orbits of Rational Maps 261

161. Further, our method can be applied to any endomorphism of a compact metric rcpace for which results on the Ruelle operator and zeta functions hold, analogous to Propositions 1.1, 5.1. One such class are piecewise monotonic expanding maps of the iiiterval, as described in [22]. In this case it should be quite straightforward to obtain 1111 analogue of our main result.

In Section 8, we describe a connection between preperiodic points of rational maps find a problem in diophantine number theory. We consider polynomial mappings, whose coefficients are algebraic integers. In this setting, preperiodic poilits have an interpretation in terms of a generalized Mahler measure. We can apply our main result 1.0 deduce an asymptotic counting formula of a number theoretic nature.

1. Subshifts of finite type

Throughout this section, we let A be a k x C, zero-one aperiodic matrix, and we

I 00 define

C,+ = {z E n{l, ..., C} : A(z,,zn+l) = 1 for all n 2 0 . n=O

For any Q E (0, l) , we can define a metric d+ on C$ by d+(z, y) = a", where n is the largest positive integer for which ti = yi for 0 5 i < n. With respect to this metric, C z is a compact, zero dimensional space. The continuous map u : C i 4 Z$ giv,en by (02)" = zn+l ie called a (one-sided) subehift of finite type. In fact, t~ is a bounded - to-one, local homeomorphism.

For g E C(C2; C ), define

Var,(g) = SUP { (g(Z) - g(y)( : ti = $/i for 0 5 i 5 n} , and for any 0 < a < 1, define a norm I . l a by

lgla = sup{- : n 2 o . I The space Fz(4:) defined by

F,+(C) = (9 E C(C,+;C) : Ig1a < 00)

is a Banach space when endowed with the norm llglla = lgla + Ilgllm, where 11 - llm is the uniform norm. Let F$(R) denote the subspace C(Cf; ;R) n F $ ( C ) of F$(C).

Two functions f , g E F$(C) are said to be cohomologous (written f - 8) if there exists a continuous function h such that f = g + h o u - h. A function f is called a coboundary if it is cohomologous to the function which is identically zero. Given g E C(Ci; lR.) , we define a real number P(g), called the pressure of g by

P(g) = sup { h(u) + 1 gdu : v is a u- invariant Bore1 probability measure . I When g E F$(R), the supremum is attained by a unique measure p = pg, (i.e., P ( g ) = h(p)+Jgdp) called the equilibrium state or Gibbs state of g. If f , g E F$(R)

262 Math. Nachr. 186 (1997)

are functions such that f - g is cohomologous to a constant function, then f , g haw the same equilibrium state.

For f E F$(G), we define the Ruelle operator L j : FZ(6:) -+ F:(G) by

where the summation is over the finite set { y E C i : a y = 2 1. The properties of thv Ruelle operator which we shall require are summarised in the following proposition. We let p ( L ) denote the spectral radius of the operator L.

Proposition 1.1. ([20] Parts (i)-(iii), [17] Part (iv)). Let f = u + i v E F$(C) h given.

(i) There is a unique simple posiiive maximal eigenvalue eP(") of L, wiih c o r m sponding strictly positive eigenfunction h = h, E F$(IR). Furiher, the remainder o/ fhe spectrum ofLu : F:(C) 4 F:(Q:) is contained in a disc of radius strictly less fhan eP(").

(ii) There as a unique probabiliiy measure u = uu such Ihai L:u, = eP(")uu. Furihei,, ihe measures put uu are related by the formula & = h,.

(iii) & 4 h,ludu, a s n -+ 00 exponentially fast, uniformly for all v E F$(G), and furthermore, J h,du, = 1.

(iv) p(Cj ) 5 eP(") and L j has an eigenvalue a = eiO+p(u), for some 0 5 (I < 27r, o/

modulus eP(") if and only i f u - a + w o a - w E C ( C 2 ; 2 u Z ) , forsome w E C(Ci ;G) . If Cj has no eigenvalues of modulus eP(") fhen p(Lj) < eP(").

We can extend the definition of pressure to functions f = u + iu E F$(C), where u - 2uM + c for some M E C(Ci; Z) and c E [0,27r), by defining P( f ) = P(u) + c. Similarly, we can define h j = la, and uj = u,. For each such f E F$(G), therr exists an open neighbourhood N ( f ) of f in F$(G), such that the maps f ep(') and f H hj have analytic extensions to N(f), such that Lthg = eP(g)hg holds for all

We define P ( g ) to be the principal branch of log(eP(g)) , for each g E N ( f ) . Also, thtl map f H uj can be extended to a weak -*-analytic map on a neighbourhood of f, (which we again denote by N( f ) ) , by g H us such that C:u, = ep(g)ug and hgdug = I hold for all g E N(f). (Weak-*-analytic means that for each w E F$(C), the map N( f ) -+ C given by g w J wdug is analytic. This extension of the definition of vj

occurs in the work of LALLEY [9].)

9 E N ( f 1.

2. Hyperbolic rational maps

Let R : G, -+ 6, be a rational map of the Riemann sphere G, = G U {co), of degree d 2 2. Let J(R) denote the Julia set of R. This is defined to be the set. of points in G cx, around which the family of forward iterations of .R is not normal in


khe sense of Monte1 (see 111, Chapter 3). The Julia set J (R) is non - empty, compact, perfect and satisfies

R(J(R) ) = J(R) = R-'(J(R)) .

Frequently, we shall consider the restriction of R to J ( R ) without special indication. We will consider only expanding rational maps. A rational map R is called expanding

if there are real numbers c > 0, X > 1 such that for any n 2 1,

The norm 11 . 11 can be the norm derived either from the Euclidean metric or from the spherical metric on (Em. Expanding rational maps are also refered to as being 'hyperbolic', and we will use these terms interchangeably.

The expanding condition leads to the existence of a Markov partition (see [20]). This is a cover of J(R) by a finite collection of rectangles B1,. .., Bk (k 2 d) . Each rectangle is a closed, non-empty set with dense interior. The rectangles have pairwise disjoint interiors, and each set R(B;) is a union of the sets (Bj)?=i. The rectangles may be chosen so that their diameters are arbitrarily small. One may then define a k x k, 0 , l matrix A by

1 if BinR- ' (B j ) # 0 , 0 otherwise.

A ( i , j ) =

Since R(J(R) is topologically exact, the subshift C i defined by A is topologically mixing or, equivalently, the matrix A is aperiodic. This means that there exists n 2 1 such that A n ( i , j ) > 0 for any 1 5 i , j _< k.

Proposition 2.1. Let R : Goo + CC ,-,, be an ezpanding rational map. There exists an aperiodic 0 - I matrix A and a map K : C: --+ J ( R ) such that ?r o u = R o ?r and

(i) a i s Lapschitz.

(ii) K is surjective.

(iii) ?r i s 1 - 1 on a set of full measure, with respeci io any ergodic, fully supporied

(iv) K i s bounded - t o - one.

The possibility of points passing through the boundaries of the rectangles (B; ) means that there is not a one- to-one correspondence between the periodic orbits of R and u. However, by simplifying an argument for hyperbolic diffeomorphisms due to MANNING 1111, it is possible to show that the set of periodic orbits which intersect Ui8R, is finite, which suffices for our purposes.

Proposition 2.1 allows us to duplicate the definition of pressure and equilibrium dates for the rational map R.

Define a function J ( R ) -+ R by

invariant measure.

t - log IR'(z)I

264 Math. Nachr. 186 (19!J7)

Since J ( R ) contains no critical points, the function is well defined and Holder contiri uous on J ( R ) . Define a map r : C: -, R by

(2.2) = log IR'(r(4)l which satisfies r > 0 and r E F$(R), for some a E (0 , l ) .

By the Bowen-Manning-McClusky formula [2], [12], the function R + R givcw by (2.3) 1 - P(-tloglR'() has a unique zero at t = 6, where 6 = dimH(J(R)) is the AausdorfT dimension of J(R) .

Define a measure m on J ( R ) by m = p 4 r o T- ' , which is equivalent to thr 6-dimensional Hausdorff measure. The Lyapunov exponent x of m is defined to be

(2-4) x = J loglR'(t.)ldm(4, J ( R )

and further, x > 0. We define another Bore1 probability measure n on J ( R ) , called the 6-conformal measure, by n = V 4 r o ?r-l. In particular, by Propositions 1.1, 2.1, there is a strictly positive function h : J(R) + R such that = h. (The propertiw of conformal measure for rational maps are studied in detail in [5].)

Definition 2.2. A point z E J ( R ) is called preperiodic if it is not periodic under R, but for some n 2 1, R"r is a periodic point. A preperiodic orbit r is defined to bv the forward orbit of a preperiodic point, that is r = { R " t } n > o , where z E J(R) is i l

preperiodic point. -

Let r be the preperiodic orbit of t E J ( R ) . Let m, p > 0 be defined by

m = inf {k 2 1 : Rkt is periodic} ,

p = inf {k 2 1 : R ' ( R ~ Z ) = R ~ Z )

For such m, p, the orbit r can be written as a union of disjoint sets, r = 71 U q , r1 n 7-2 = 0, where

and

and 71 = { z , Rz, R2z, . . . , Rrn-lz},

r2 = { R ~ Z , R~+'Z, . . . , ~ " + p - ' z } .

We denote this decomposition by r = 71 * 72. Let A1(r) = m and A z ( r ) = p , and lcl. A ( T ) be the length of r, defined by

A(r) = A l ( 7 ) + A2(.).

Now define a weight R(r) on r by

Widdington, Preperiodic Orbits of Rational Maps 265

Ily the Chain Rule, we have that

(2.6) W.r) = %(T)%(T) 1

nrtd

For a periodic orbit (, of least period kl we define

(2.7) P ( t ) = I(R"'(Y)l for any Y E t . The definition of preperiodic orbit carries over naturally to subshifts of finite type

(or to any endomorphism for that matter), and we shall use this fact without further (axplanation. In the case of shifts, we define the weight R(7) by

where 7 denotes the preperiodic orbit of a point 2 E C i , and 47) denotes the length of 7.

3. A classification of rational maps

We now describe a condition which partitions hyperbolic rational maps into two disjoint classes.

Definition 3.1. Consider the equation

(3.1) w ( ~ ( z ) ) = I R ' ( L ) ~ ' " w ( L ) for L E J ( R ) ,

where w E C(J(R); Sl) and a E IR. A rational map is of f y p e (I) or type (11) according to the following conditions:

constant, (called trivial solutions).

(I) Equation (3.1) has no solutions except a = 0 and w

(11) There exists a non-trivial solution of (3.1).

Remark 3.2. (i) Equation (3.1) has an equivalent formulation in terms of symbolic dynamics; namely

where 6 E C(C;;S') and a E R. After substituting G for w , conditions (I) and (11) have an identical formulation and are equivalent to those stated for R. We conclude from [16], Proposition 6.4, that any non- trivial solution a of (3.1) is isolated. Thus there exists a least positive solution ir. Furthermore, any other solution a of (3.1) is of the form a = Lir, for some k E P.

~ ( u z ) = e'ar(s)~(z) for t E

266 Math. Nachr. 186 (19!r7)

(ii) By ergodicity, there is no non-trivial solution of (3.1) with a = 0 and w noii

(iii) Any hyperbolic rational map is of type (I) or type (11), but not both. (iv) Conditions (11) are analogous to an Anosov flow being weak mixing or ~ i t d

(v) The above classification could be extended to rational maps which are iiol

constant.

weak mixing respectively.

hyperbolic, but it is not clear what its significance would be in that case.

In order to show that our classification is well defined, we require the followiri~ lemma.

Lemma 3.3. Classes (I), (11) are invariant under conjugalion by M6bius imnsjot~- maiions.

Proof. Let R be a hyperbolic rational map and let T = gRg-', where g : Cm + (En, is a Mobius map. In particular g l J ( R ) : J ( R ) -+ J(T) is a bijection. Differentiation id the conjugacy relation gives

and by substituting into (3.1), we obtain

Hence, we may write

(3.2) P V w ) = IT'(w,l'"P(w, ? w E J ( T ) 7

where fi : J(T) --t S" is given by

P(w) = (w 0 9-1 * 1st 0 9-1 li.) ( w ) .

Thus (3.1), (3.2) imply that if R is of type (I) or type (11), then T is also of type (I) or ( I I ) , respectively. (By Remark 3.2(ii), we do not need to check that 0 is nor] constant). L I

Let C denote the class of all continuous maps C, + Cm, and let R denote 11iv subset of all rational functions. There is a natural metric p on C given by

p(R,S) = sup{G(Rz,Sz) : z E C,}

where 6 is the spherical metric on Cm. With this topology, R is a closed subset of C Define the subset of R by

%,q = { R E R : R = - i n r e d u c e d f o r m , P d e g P L p , d e g Q s q Q


r i i i t l let R d denote R d , d , that is the set of rational maps of degree less than or equal to d. Define a continuous map aP,$ : R2(p+q+2)\(O} --* Ft,,$ by

I d l b be the Lebesgue measure on Rk. We may define a measure Ap,q on %,( by pushing forward Lebesgue measure under ap,q, namely

Again we denote @d,d and Xd,d by @d and Ad, respectively. Let H d = { R E R d : R is hyperbolic}, which is open in R d , [7]. Since it is also

lion - empty, H d has positive Ad measure. Finally, we remark that maps of degree d liave full Ad measure in R d .

Proposition 3.4. A l m o s i all m a p s an H d are of type (I) , with respect l o the measure A d .

Proof. Let R(z) = be a rational map of degree d, where

D a

i=O j =O

and max{p, q } = d. By Corollary 9.4 in [13], R has a fixed point 10 E J(R) . By replacing R(z) by

/<(I + 10) - zo, we may suppose that R has a fixed point at I = 0. Thus, we may write / - ' ( I ) = I U ( Z ) , for a polynomial U of degree less than or equal to d - 1, and regard our parameter space as R d - l , d .

We may also assume that U ( 0 ) # 0, as all maps with U ( 0 ) = 0 have zero & l , d iiieasure. By dividing U, Q by U(O) , we may suppose that U ( 0 ) = 1, and regard R as king chosen from R d - z , d . Evidently, we have that

I f R is of type (II), then by Remark l(i), there is a least positive real number a such I-hat IR'(0)l'" = 1, and hence IqOl2'" = 1. Thus

q o E U { I E C : 111=e?} ,

which is a countable union of concentric circles. The two dimensional Lebesgue measure of this set is zero. Thus the Ad-2,d measure of maps of type (11) in Rd-2,d is zero. Hence it is also of zero Ad measure in R d .

Finally, the hyperbolic maps of type (11) have zero Ad measure in H d , as required.

m c Z

0

268 Math. N d r . 186 ( l ! ~ ? )

Remark 3.5. Examples of maps of type (11) are given by maps of the form R(t ) .. czd for d 1 2 and c E Q: \ (0). Other than these rather simple examples (and tlwrr conjugates under Mobius transformations), we know of no other maps of type (11).

4. The zeta function

We now study a class of zeta functions for subshifts of finite type. We begin with some more notation. For f E F$(lR.), we define

For each n 1 1 let Fix(u") denote the set of fixed points of 6". For n 2 2, definc I I

set E(n) C C: by

E(n) = { y such that there exist p, m with n = m + p ,

(4.1) for some 1 5 p, m 5 n - 1 such that uJy is not

periodic for 0 5 j 5 rn - 1, and amy E Fix(aP)} . For s E Q: , we define formally a ceta function &, by

M

The function c0(s) is well defined and analytic whenever the infinite series converge8 In order to analyse Cu(s), we will examine a more general zeta function, defined by

(4.3) m

for g E DC(f)> where f E F$(Q:) and

for some E > 0.

Lemma 4.1. For any n 2 2, ihe following ideniiiy holds

n- I

Recently, the author haa obbined a more complete description of hypubolic mrp of types ( I ) md (1% P31

Wtrddington, Preperiodic Orbits of Rational Maps

I'roof. By definition, we may write E(n) as a disjoint union

n-1

E(n) = u F(n,p) p= 1

where

F ( n , p ) = { y : a"-P~EFix(uP),d(y) $Fix(uP)forallO< j s n - p - 1 }

Kvidently, we also have that if un-Py E Fix(@), then

a J ( y ) $ Fix(uP) for all 0 5 j 5 n - p - 1

i f and only if ~ ~ - ~ - 1 # yn-1, and hence

F(n,p) = { y : amy E Fix(aP), ~ " - ~ - ' y @ Fix(uP)} .

The result then follows by summation.

For f E F$(C) and g E N ( f ) , we define a number C(g) by

(1.4) C(g) = ep( , ) [ h, d p , - e'h, d p , . J J

Virst we remark that the map N ( f ) Q: given by g H

Lemma 4.2. T h e number C(g) may be ezpressed as

Proof. The proof is the following formal calculation,

(4.5)

where we have used Proposition 1.1 repeatedly.

269

0

Proposition 4.3. For any f E F$(IR), the following siaiemenis hold: (i) C(f) > 0.

270 Math. Nachr. 186 (I!J!l7\

(ii) If E > 0 i s sufficienily small ihen C(g) # 0 for all g E D, (f).

Proof . (i) Since h j > 0 by Proposition l . l ( i ) , there. is a constant c > 0 such t , h i r l hj > C. Thus

It suffices to show that there is an t o E {1,2, . . . , d } for which there exid i , j E { 1,2, . . . , d} with A(i , zo)A(j, 20) # 0, or equivalently A(i, 20) = 1 = A( j , x,,) If no such t o there exists, then each column of A contains at most one 1. Heiiw either A has a row of O's, which is immediately ruled out by aperiodicity, or else A i M a permutation matrix. However, if A is a permutation matrix then Akn = I for SOIIII*

n 2 1 and all t = 1 , 2 , . . . . Thus again A is not aperiodic. Thus C(f) > 0 as requircvl (ii) This follows from (i) and the fact that g ++ C(g) is analytic, and hence contimi

ous. I I

5. Extending the zeta function

In this section, we prove a general result about the zeta function Z(g) defined i t 1

(4.3). The results of this section are analogous to those proved in Chapter 5 of [ I f \ ] for the periodic point zeta function.

For f E F,$(C), define formally a zeta function dl(f) by

The analytic domain of 4l(f) is described by the following proposition.

Proposition 5.1. ([15] Theorem 1, [17] Theorem 3.) Let f E F$(C). (a) I fP(Ref) < 0 o r t fP(Ref) = 0 and p ( L j ) < 1 then fhere ezisls E > 0 such I l i r r l

41(g) i s analytic and non - zero in Dc(f). (b) IfP(Ref) = 0 and p (L j ) = 1, o r equivalently f - Re(f )+ia+Ss iM, then Ihwi

ezasts E > O such that (Pl(g) as analytic and non-zero in Dc(f) \ {g : P(g) = 0}, and s a 2 isfies

where Zl(g) i s analytic in Dc(f).

Now we define another zeta function 4 2 formally by


t o r f , w E Fz(Q: ) .

Proposition 5.2. Let f E F$(Q:). (a) IfP(Ref) < 0 or i f P ( R e f ) = 0 and p ( C j ) < 1 fhen fhere ezists E > 0 such that

(b) If P(Ref) = 0 and p(Cj ) = 1, or equivalently f - Re(f)+ia+P.lriM, then there I/ll(g) i s analytic in Dc(f).

1-xists E > 0 such that &?(g) is analytic in Dc(f) \ {g : P(g) = 0}, and satisfies

where Z2(g) is analytic in Dc(f).

Proof. The argument is essentially contained in [15], Section 4, so we include only

For s E Q: such that 181 is small, we consider the function ti sketch proof.

li%,s) = h ( s + s w ) .

Part (a) follows directly from Proposition 5.l(a) by logarithmic differentiation of tl,(g,s) at s = 0. For part (b), we apply Proposition 5.l(b) to deduce that &(g) ib: analytic in Dc(f) \ {g : P ( g ) = 0}, and satisfies

where &(g) is analytic in D,(f). Here we have used the standard formula for the 0 derivative of pressure (161, Proposition 4.10.

We can now state our first main result.

Theorem 5.3. Let f E Fz(Q: ) . (a) IfP(Ref) < 0 or ifP(Ref) = 0 and p ( C j ) < 1 fhen ihere en’sts E > 0 such fhat

%(g) i s analyiic in Dc(f). (b) If P(Ref) = 0 and ~ ( C J ) = 1, or equivalently f - Re(f) + ia + 2uiM, fhen

[here et is ts E > 0 such that Z(g) is analyiic in DE(f) \ {g : P(g) = 0 } , and may be rxvressed as

where C(g), &(g) and Z4(g) are analytic in Dc(f).

Proof. We first prove (b) and then indicate the modifications required to this ar-

(b) For E > 0 so small that Dc(f) C N ( f ) , by Proposition 1.1 we may write gument to prove (a).

(5.1)

272 Math. Nachr. 186 (l!)!U)

for any n 2 1, w E Fz(C) and g E Dc(f), where

lim lllQn1116 5 p < eRe p ( J ) = I (5.2) n-rm

with p depending only on f . For each n 2 1, the operator Qn : F$(d:) -+ F$(C) is lh projection operator corresponding to the remainder of the apectrum of &3. The nortii 111 . 111 is the operator norm on F$(C). We may suppose by standard Perturbahill Theory arguments, (see [8] or Appendix 4 of [IS]), that /3 depends only on f .

First of all note that

c c egm(w)+gp(o) r ~ F i x ( u ? ) u”’v=z

um-lwCFix(up)

(5.3)

We now consider the first term on the right-hand side of (5.3). Note that

(5.4)

where zp-lt = ~ ~ - 1 t 0 2 1 2 2 . . . . Since g and g o D have the same equilibrium stabc., we can apply Proposition 5.2 to the final two terms in (5.4), to give

(5.5)

Wntldington, Preperiodic Orbits of Rational Maps 273

I'IW all g E Dc(f) \ { g : P(g) = 0}, where Za(g) is analytic in Dc(f). Now we consider the second term on the right-hand side of (5.3). For this we have

I l y applying Proposition 5.2 again, we obtain

1 i owever, by (5.2) ,

,siiice p < 1. Thus we have

l o r all g E D c ( f ) \ { g : P ( g ) = 0}, where &(g) is analytic in Dc(f). (The value of F > 0 is given by Proposition 5.2).

By Lemma 4.1, we have the following formal identity

I3y taking

tlie result follows from (5.9), (5.3), (5.5) and (5.8). This completes the proof of part ZS(9) = z5(9) + Z6(9)1

(1 ) ) .

274 Math. Nachr. 186 (1!)!17)

(a) First suppose that P(Re f ) < 0. Choose E > 0 so t~hat P(Re g) < 0 for nll g E Dc(f). By Lemma 4.1 and (5.3)'

(5.10)

for a constant I< > 0.

such that if p > A41 then By Proposition 5.1 in [16], since P(Re f ) < 0, we may choose €1 E (0 , l ) and MI > 1)

(5.11)

Secondly, since p(L,) 5 P ( b g) < 0 for g E Dc(f), we may choose €2 f (0'1) ant1 Mz > 0 so that if n - p > M2 then

II,ty-P--' (5.12) # 111, < € 2 -

Let E' = max{sl,e2}. Then we have from (5.10)-(5.12) that

< E'

< 1 .

Thus, if P(Re f ) < 0, then Z(g) is analytic in DL(f). In the case that P(Ref) = 0 and ~ ( L c , ) < 1, we have

L ~ w = Qnw

for any n 2 1, w E F,$(C) and g E DC(f), where Qn satisfies (5.2). We can follow steps (5.3) and (5.6) to deduce that c , S r n ( W ) + 9 P ( ~ ) c

rfFix(op) umw=z u m-1 w t Fix( u p )


Irom (5.2), we have that m

The result then follows from (5.9) and Proposition 5.2(a). This completes the proof of part (a). D

6. Counting theorems

In this section, we will apply Theorem 5.3 to the zeta function CU(s) defined in (4.2). This will allow us to deduce asymptotic formulae for preperiodic orbits. We will asmme throughout that R is a hyperbolic rational map of type (I).

Proposition 6.1. The function Cu (s) saiisfies

c B Cu(s) = - + - + d)I(S) , (s - 6)2 s - 6

where ~I(s) is analyiic in an open neighbourhood of { s : R,e(s) 2 a}, B is a consfani, and C is a posiiive consiant given b y

C(-br) c = 2 ’ (.f dC1-6r)

Proof. By Theorem 5.3(a), G(s) is analytic for P(R,e(-sr)) < 0, i.e., R,e(s) > 6, by the Bowen - Manning- McClusky formula. When s = 6 + ia, (,(s) has an analytic extension to a neighbourhood of s, provided L+a+ia)r does not have 1 as an eigenvalue, by Theorem 5.3(i), (ii). When 1 is an eigenvalue of L-(b+ia)r , b ( s ) has an analytic extension to

Dc(6 + ia) \ {s E CC : P(-sr) = 0).

An argument on page 65 of [16] shows that 6 + ia is not an accumulation point of {s : P(-sr) = 0). Further, LC-(6+ia)r has 1 as an eigenvalue if an only if ar - 2n-M for M E C(C2; Z) and a E R. This holds if and only if a = 0 and M 0 since R is of type (I). Thus s = 6 + ia has a pole-free open neighbourhood if and only if a # 0.

S#&&-note that

276 Math. Nachr. 186 (1997)

where we have used the formula for the first derivative of pressure stated in Propositioii 4.10 of [16]. The value of the constant B can be calculated in a similar way, but is ot

I 1 no significance for our purposes.

Now by Lemma 4.1, we have

by substituting n = p + m. Further, we have that

m

where the manipulations make sense in the region where c u ( s ) is absolutely convergent. The final two summations in (6.1) require some further explanation. The sum over TZ is a sum over all periodic orbits of R. For a fixed z E 7 2 , the summation over y1 (z) is a sum over all 71 such that 7 is a preperiodic orbit, of the point z E J ( R ) say, with decomposition y = 71 * 7 2 and satisfying Pl(y)(z) = 6.

We now split the sum in (6.1) into two parts. Firstly, we consider the function

. . al(71 * 72)-'R2(71 * 72)-' =cc Ta "E%r1=71(r) c R2(7l * 72)' - 1

corresponding to k 2 2. We require the following proposition describing p l ( s ) .


Proposition 6.2. There ezisls an open neighbourhood U of { s : Re(s) 2 6) \ (6) such fhat i fs E U then Ipl(s)I < 03. Furiher, there etisfsE > 0 such that ifIs-61 < E

/hen I(. - 6)pl(s)( < 00.

I\', > 0 such that if u > 6 - E then Proof. Let u = Fte(s), and note that if 6 > 0 is sufficiently small, there is a constant

(6.3) P(€)" i: Kl(P(€Y - 1) 1

where ( denotes a generic periodic orbit of u, and P is defined in (2.7). Next, we have from (6.2) that

(6.4)

First note that for any fixed z E Fix(uP) the function m

satisfies

for a constant K2 > 0

278 Math. Nachr. 186 (1997)

Using Proposition 1.1, it is a straightforward exercise to show that

where $2(s) is analytic in an open neighbourhood of {s : &(s) 2 6). (This uses thc fact that R is of type (I)). Thus the conclusions of the proposition apply to pz(s).

Secondly, note that if u > 6/2, then

by [16], Proposition 5.1 and the strict monotonicity of pressure. Thus the series

M

converges for u > 612. Combining our observations about p2, p4 with (6.4) proves the proposition. 0

We now consider the remaining part of ((s), namely

Fkom (2.6), we have that

T

where the sum is over all preperiodic orbits 7 of u. We can then rewrite q ( s ) as a Stieltjes integral of the form

roo q ( s ) = J t- 'dcr(t)

0

where a ( t ) = #{7 : R ( y ) 5 2 ) .

proposition. Since cu(s ) = ~ l ( s ) + p 1 ( s ) , we can combine Propositions 6.1, 6.2 to give the following

Proposition 6.3. The function q(s) safisfies

where $s(s) is analyiic in an open neighbourhood of {s : Re(5) 2 a}, C > 0 is llr.r, consiani given in Proposition 6.1, and D is an unspecified consiant.


We now require a Tauberian theorem due to DELANGE.

Proposition 6.4. ([4] .) Suppose that

holds f o r %(s) > A, where el(s), ~ 4 s ) are analytic in an open neighbourhood of { fi : Re(s) 2 A}, and el (A) # 0 . Then

a(t) - el(A)txlogt as I t -, 00.

By combining Propositions 6.3, 6.4 we have that

(6.7) = #{7 : 7q7) 5 t } - c t 6 i o g t t - oo where S = dimH(J(R)) and C is given by Proposition 6.1. It is important to recall I.liat in (6.7), y denotes a generic preperiodic orbit of the shift map u modelling R.

We now define P( t ) = # { r : ‘R(7) 5 tl

where, as usual, r denotes a generic preperiodic orbit of R. It remains to compare r r ( t ) and p(t). As described in Section 2, the periodic orbits of R and u differ by only liriite number of orbits, say 7(’), 7(2), . . . , $’) of Q.

From (6.6), we may write

- - where

T,( t ) = # { y : er”(y) 5 t , umy = t for some m 2 0} . 13y applying the Wiener -1kehara Tauberian Theorem ([lo]) to (6.6), we deduce that

(6.8)

Math. Nachr. 186 (19'17) 280

and furthermore, EL, C.tE.y(') Wi)

t 6

is bounded as t -+ 00 by (6.8). From this and (6.7) we deduce that

(6.10) P( t ) - CPlog t 88 t - 00.

Finally we have from Proposition 2.1 and (4.4) that

1 C = - C(-br ) X2

h( 1 - lR'1'6) dm . = -J 1 X 2

Our main result now follows immediately from (6.10) and (6.11).

Theorem 6.5. Suppose ihaf R is a hyperbolic raiional m a p of f y p e (I). T h e n

j { T : R(7) _< t } - CtJlogt as t - 00,

where 6 = dimH(J(R)) and C > 0 is a constani given by

C = -/ 1 h(1- lR'l-6) dm. X2

7. The zeta function for R

Our analysis has avoided direct use of the zeta function ( ~ ( 8 ) . This was purely for notational convenience. The analytic domain of { ~ ( s ) is of some interest in its ow11 right, so we prove the following analogue of Proposition 6.1.

Theorem 7.1. Suppose R i s of type (I). Then the function CR(S) safisf ies

where 6 = dimH(J(R)), C > O is a consfanf given by Theorem 6.5, D is an unspecified constant and +4(s) is analytic in an open neighbourhood of {s : Re(s) 2 6).

finite number of orbits, say y('), ~ ( ~ 1 , . . . , y(') of u. Proof. As described in Section 2, the periodic orbits of R and u differ by only ;I

Witddington, Preperiodic Orbits of Rational Maps 281

13y applying the steps in (6.1) to ( ~ ( s ) , we may write

where

Firstly note that m

p(7(i))-8k < c A - W r P < 00 I - k = l

provided that Re(s) > 0, where X > 1 is given by (6.9). Secondly we have

I

P 6 ( s ) = c c c 'Rl(71 *7(i))-J i=l oE7(t) 71=71(r)

is analytk in U \ {6}, where U is an open neighbourhood of {s : Re(s) 2 6). hr ther- inore, there exists E > 0 such that if Is - a( < E then l(s - 6)ps(s)I < 00. (This follows by duplicating the argument used to estimate p2(s) in Proposition 6.2). Thus, p5(5)

satisfies the two properties just stated for ps(s). The theorem then follows from (7.1) and Proposition 6.1. 0

8. Diophantine properties of preperiodic points

A classical problem in number theory is the following question. Let D be a compact subset of a . Which algebraic integers belong to D, together with all their conjugates?

In the case that D = D(0; 1) is the closed unit disc, if z is a point with this property, (,hen z = 0 or z is a root of unity. This is a classical result, due to KRONECKER. The general problem for an arbitrary domain D remains unsolved.

Let P be a polynomial of degree d , at least two. Its filled in Julia set, K = K ( P ) , is defined to be the complement of the basin of attraction of the point at infinity, under I'. First we give two results, due to MOUSSA [14], which solve the problem when 11 = K ( P ) , for certain polynomials P.

Let 8 be an algebraic integer of degree I , and let 81 = 8,&, . . . , 81 be its conjugates. Let A[8] denote the set of linear combinations ~ ~ l ~ p j 6 ' J 1 where the p , L are rational

282 Math. Nachr. 186 (1997)

Coefficients, which forms a ring. We define the conjugate rings A[Bi], i = 2,3,. . . , I , it) the same way.

First note that if z E A[@] , then z itself is an algebraic integer of degree at most. I , the conjugates of which belong to the corresponding A[&].

We say that z is an algebraic integer over A[@] if z is a solution of an equatioii Q ( r ) = 0, where Q is a monic polynomial with coefficients in A[B]. Let Z[B] be the SVI

of such algebraic integers. If Q is an irreducible polynomial over A[8], the degree of Q is equal to the degrw

of the algebraic integer z over A[O]. The other roots of Q are the conjugates of z. WV shall call these internal conjugates. By conjugating the coefficients of Q , we obtain polynomials Qil for i = 1 , 2 , . . . , I ,

where each Qi has coefficients in A[Bi]. Since nf,, Qi is a polynomial with algebraic integer coefficients, z is also an algebraic integer of degree 1. deg Q, and therefore ha< ( I - 1). deg Q other conjugates, which are solutions of Qi(w) = 0, for some 1 5 i 5 1. We shall call the latter external conjugates.

Let P be a polynomial with coefficients in A[B]. By conjugating its coefficients, wv may define conjugate polynomials P = PI , P2, . . . , PI.

Proposition 8.1. ([14].) Let P be a monic polynomial wiih coeficients in A[O]. Then z has finite forward orbii under P if and only if all the following conditio1t.r hold

(i) z E @". (ii) z and all i t s internal conjugafes belong i o K ( P ) .

(iii) The etiernal conjugates of z belong to the corresponding K(P i ) .

Points with finite forward orbit are precisely those points which are either periodic or preperiodic.

A convenient way to express the conditions in Proposition 8.1 is to define an ex. tension of the usual Mahler measure. Let P be a monic polynomial of degree d 2 2. Define P p : Q: + IR by

(The limit in this definition always exists). In particular, p p ( z ) > 1 if and only il' z 4 K ( P ) . Given an algebraic integer z , with conjugates z1 = z , 2 2 , . . . , z,, we defiiw its generalized Mahler measure M p ( z ) by

r

i=l

If P ( z ) = z 2 , then /3p(z) = sup{lzI,l}, and we recover the usual Mahier measiiw Proposition 8.1 can be restated as follows.

Proposition 8.2. ([14].) Let P be a monic polynomial with coeficienis in A[()] Then z has finite forward orbii if and only if z E I(@] and M p ( z ) = I .

Wtrddington, Preperiodic Orbits of Rational Maps 283

We can now apply our results to count algebraic integers z over A[B] with Mahler itic-itsure M p ( z ) = 1, for which a growth condition on the modulus of the derivative of I’ i s satisfied.

Theorem 8.3. Let P be a monic polynomial of degree d 2 2 , with coefictents in l I [ O ] . Suppose P i s hyperbolic of type (I). Then

vliore ihe values of6, C are as in Theorem 6 .5 , and N ( z ) i s defined as in the Zntro- tirction.

Remark 8.4. (i) A similar formula holds for maps of type (11). (ii) There are many example of hyperbolic rational maps, whose coefficients are

ilgebraic integers.

I-’ r o o f of Theorem 8.3. To apply Theorem 6.5 to the situation described in Prop- ition 8.2, we only have to note that

g{z : z periodic, ( (PN(*)) ‘ ( z )I 5 t } - C’t6 as t - 0 0 ,

in a constant C’ > 0. 0

\ cknowledgement s

I wish to thank Professor M. DENKER for some useful comments and references during kr course of this work. I also wish to thank the SFB 170, Gottingen, for its hospitality and tipport. My interest in complez dynamics originated in an interesting series of lectures given 11 Profearor A.F. BEARDON as part of the 3rd International Summer School, held at the ‘triuersity of Jyvaskyla, Finland, during August, 1993.

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Insiiiui fir Maihemotirche Siochasiik Loiresir. 13 D - 37083 Giiiingcn c -mail: [email protected]

zeta functions and asymptotic formulae for preperiodic orbits of hyperbolic rational maps

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