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Canonical Thurston Obstructions
Advances in Mathematics 158, 154168 (2001) doi:10.1006aima.2000.1971, available online at http:www.idealibrary.com on
Canonical Thurston Obstructions Kevin M. Pilgrim 1 Department of Mathematics and Statistics, University of Missouri at Rolla, Rolla, Missouri 65409-0020 E-mail: pilgrimumr.edu Communicated by R. D. Mauldin Received October 19, 1999; accepted October 25, 2000
We refine Douady and Hubbard's proof of Thurston's topological characterization of rational functions by proving the following theorem. Let f: S 2 Ä S 2 be a branched covering with finite postcritical set P f and hyperbolic orbifold. Let 1 c denote the set of all homotopy classes # of nonperipheral, simple closed curves in S 2 &P f such that the length of the unique geodesic homotopic to # tends to zero under iteration of the Thurston map induced by f on Teichmuller space. Then either 1 c is empty, and f is equivalent to a rational function, or else 1 c is a Thurston obstruction. 2001 Academic Press
1. INTRODUCTION For relevant background we refer the reader to [1]. Let f: S 2 Ä S 2 be an orientation-preserving branched covering map of degree d 2, and let P f = n>0 f b n (0 f ) where 0 f is the set of critical points of f. We say f is a Thurston map if P f is finite; we assume this throughout this work. Two Thurston maps f, g are combinatorially equivalent if there are homeomorphisms %, %$: (S 2, P f ) Ä (S 2, P g ) such that % is isotopic to %$ relative to P f and % b f =g b %$. The orbifold Of associated to f is the topological orbifold with underlying space S 2 and whose weight &(x) at x is the least common multiple of the local degree of f over all iterated preimages of x. The orbifold Of is said to be hyperbolic if the Euler characteristic /(Of )=2& x # Pf (1&1&(x)) is negative. A simple closed curve # # S 2 &P f is nonperipheral if each component of 2 S &# contains at least two points of P f . A nonempty set 1=[# 1 , ..., # n ] of simple, closed, nonperipheral, non-homotopic curves on S 2 &P f we call a multicurve; it is said to be f-stable if for any # # 1, all components of 1
Research partially supported by NSF Grant DMS 9996070.
154 0001-870801 35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.
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f &1 (#) are either peripheral, or homotopic to an element of 1. Associated to an f-stable multicurve 1 is a linear map f1 : R 1 Ä R 1 defined as follows. Let # i, j, : be the components of f &1 (# j ) which are homotopic to # i in S 2 &P f . Define 1 #i , d i, : i, j, :
f 1 (# j )=:
where d i, j, : is the (positive) degree of the map f | # i, j, : : # i, j, : Ä # j . Then f 1 has spectral radius realized by a real nonnegative eigenvalue *( f, 1 ). Thurston's characterization is Theorem (Thurston's Characterization and Rigidity Theorem). A Thurston map f with hyperbolic orbifold is equivalent to a rational function if and only if for any f-stable multicurve 1 we have *( f, 1 )<1. In that case, the rational function is unique up to conjugation by an automorphism of the Riemann sphere. If Of is hyperbolic and 1 is an f-stable multicurve with *( f, 1 ) 1 we will call 1 a Thurston obstruction. This is something of a misnomer; in Section 2 we introduce the idea of a simple Thurston obstruction which is better suited for our purposes. The idea of the proof is the following. Associated to f is a Teichmuller space Tf modelled on (S 2, P f ), and an analytic self-map _ f : Tf Ä Tf . The existence of a rational map combinatorially equivalent to f is equivalent to the existence of a fixed point of _ f . The map _ f is distance-nonincreasing for the Teichmuller metric, and if the associated orbifold is hyperbolic, _ 2f decreases distances, though not necessarily uniformly. To find a fixed point, one chooses arbitrarily { 0 # Tf and considers the sequence { i =_ fb i ({ 0 ). If [{ i ] fails to converge, then the length of the shortest geodesic on { i , in its natural hyperbolic metric, must become arbitrarily small. In this case, for some i sufficiently large, the family of geodesics on { i which are both sufficiently short and sufficiently shorter than any other geodesics on { i form a Thurston obstruction. For a nonperipheral simple closed curve #/S 2 &P f let l { (#) denote the hyperbolic length of the unique geodesic on the marked Riemann surface given by { which is homotopic to #. In [1] the authors show by example that it is possible for curves of two different obstructions to intersect, thus preventing their lengths from becoming simultaneously small. Hence, if 1 is an obstruction and # # 1, then it is not necessarily true that inf i [l {i (#)]= 0. Moreover, their proof does not explicitly show that if f is obstructed,
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then inf i [l {i (#)]=0 for some fixed curve #. Thus it is conceivable that, for each i, there is a curve # i such that inf[l {i (# i )]=0 i
while for fixed i inf[l {j (# i )]>0. j
We will rule out this possibility, and in so doing establish Theorem 1.1 (Canonical Obstruction). Let f be a Thurston map with hyperbolic orbifold, and let 1 c denote the set of all homotopy classes of nonperipheral, simple closed curves # in S 2 &P f such that l {i (#) Ä 0 as i Ä . (1)
If 1 c is empty, then f is combinatorially equivalent to a rational
map. (2) Otherwise, 1 c is an f-stable multicurve for which *( f, 1 c ) 1, and hence is a canonically defined Thurston obstruction to the existence of a rational map combinatorially equivalent to f. The proof also shows, with the same hypotheses, Theorem 1.2 (Curves Degenerate or Stay Bounded). Let # be a nonperipheral, simple closed curve in S 2 &P f . (1)
If # # 1 c , then l {i (#) Ä 0 as i Ä .
(2) If # Â 1 c , then l {i (#) E for all i, where E is a positive constant depending on { 0 but not on #. Remarks. (1) In fact, 1 c has the additional property of being simple (see Subsection 2.2) and is also disjoint from all other simple obstructions. The idea of non-simple is the following. To an obstruction, add a disjoint invariant multicurve which is not an obstruction. The added curves are in a sense extraneous. A simple obstruction contains no such extraneous curves. (2) Our arguments show that 1 c equals the set of curves which become arbitrarily short. (3) It is tempting to view the obstruction 1 c as the analog, for branched coverings, of the tori in the canonical torus decomposition for irreducible three-manifolds; this is the subject of work in progress. (4) We do not know if _ f extends in a reasonable way to one of the natural compactifications of Tf .
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(5) Our proof uses no analytic tools other than those in [1] and is based mostly on combinatorial refinements of their arguments. Outline of This Paper. In Section 2 we introduce our combinatorial refinements and reduce the proof of Theorem 1.1 to Proposition 2.7, Short Curves Remain Short, which we prove in Section 3.
2. BACKGROUND 2.1. Nonnegative Matrices We shall need some basic facts from the theory of nonnegative square matrices; see [2, Vol. 2, Chap. XIII]. Such a matrix A always has a largest nonnegative (or leading) eigenvalue *(A) equal to its spectral radius. Irreducible Matrices. A nonnegative n-by-n matrix A is called irreducible if no permutation of the indices places it in block lower-triangular form. In this case, given i, j there exists q with 0 q n such that (Aq ) i, j {0. By the Frobenius Theorem, A has a positive eigenvalue *(A) equal to its spectral radius, and up to scale, there is exactly one positive eigenvector corresponding to *(A). Reducible Matrices. By applying a permutation of the indices, A may be assumed to be in the form
\
A 11 A 21 }}} As1
0 A 22 }}} As2
}}} }}} }}} }}}
0 0 , 0 A ss
+
where the blocks A jj are irreducible. Moreover, *(A)=sup *(A jj ). j
2.2. Multicurves We shall need to consider transformations for multicurves 1 which are not necessarily f-stable. Define f 1 : R 1 Ä R 1 by the same formula. We say 1 is irreducible if the matrix ( f 1 ) is irreducible. Let 1 be a multicurve and A=( f 1 ). Let 1 j denote the set of indices (curves in 1) corresponding to the jth block in the decomposition given in the previous subsection, so that A jj =( f 1j ). We call the 1 j 's the irreducible components of 1. As sets, they are well-defined up to permutations of the indices j.
158 Depth. set
KEVIN M. PILGRIM
Let 1 be a multicurve, A=( f 1 ), and suppose *(A) 1. Then the
1 $= .
1j
*(Ajj ) 1
is nonempty. Definition 2.1 (Depth, Simple Obstruction). Let # # 1. The depth of # with respect to 1 is the smallest integer k 0 for which there exists #$ # 1 $ with (A k ) ##$ {0. If no such k exists, we say # has infinite depth and call # extraneous with respect to 1. The set of elements of 1 of finite and infinite depth we denote by 1 2 and 1 , respectively. A Thurston obstruction 1 is called simple if 1=1 2 . Thus, the depth is how many backwards iterates are necessary to make # equal to the preimage of some curve #$ # 1 $. In particular, curves of depth zero lie in 1 $. Note that the depth of a curve depends on the ambient multicurve and may change if curves are added or subtracted. Proposition 2.1. If 1 is a Thurston obstruction, then 1 2 is a simple Thurston obstruction. In particular, after applying a permutation, ( f 1 )=
f 1
\C
0 , f 12
+
where *( f 1 )<1 and *( f 12 )=*( f 1 ) 1. Proof. A preimage of a curve of finite depth is of finite depth, so 1 2 is f-stable and the block-lower-triangular form holds. Furthermore, the set [1 j ] of irreducible components of 1 is the disjoint union of the set of irreducible components for 1 and 1 2 . So *( f 1j )<1 for all 1 j /1 and *( f 1j )=*( f 1 ) for some 1 j /1 2 . Hence *( f 1 )<1 and *( f 12 )=*( f 1 ). K Finiteness Results. A generalization on [1, Lemma 1.2] is that the set of possible transformations f 1 depends only on the degree d=deg f and the cardinality p of P f . As consequences, we have Proposition 2.2. There is a number ; depending only on d, p with 0< ; 1 such that if 1 is any irreducible multicurve for which *( f 1 ) 1, and if v is the unique positive eigenvector of f 1 with sup norm equal to 1, then the smallest coordinate of v is bounded below by ;.
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Proposition 2.3. There is a positive integer 2 depending only on p, d such that whenever # has finite depth with respect to some 1, then its depth is less than or equal to 2. In fact, 2 p&4. 2.3. Quasi-Nondecreasing Sequences Definition 2.2. Let } # R. A sequence [x i ] i=0 of real numbers is called }-quasi-nondecreasing if for all i 1
If [x i ] is quasi-nondecreasing and unbounded then x i Ä + .
2.4. Notation and Basic Facts Throughout, we will use the notation and basic results of [1]. In particular: v p=*P f , d=deg f. v {, an element of Tf . v w(#, {)=&log l { (#). v L { =[w(#, {)]/R, where # ranges over all nonperipheral simple closed curves in S 2 &P f . v w({)=sup #[w(#, {)]. v w(1, {)=sup # # 1 [w(#, {)]. v A=&log log(- 2+1)=0.126..., the smallest value of w(#, {) guaranteeing that the geodesic homotopic to # on { is simple and disjoint from any other geodesic #$ with w(#$, {) A. Thus for any { there are at most p&3 curves # with w(#, {) A. v For J>0 let ]a, b[ be the lowest interval in R&L { of length J, where a A, and define 1 J, { =[# | w(#, {) b].
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Thus, 1 J, { is the multicurve consisting of the set of all geodesics on { which are both sufficiently short and sufficiently shorter than any other curves. Note that if B(J)=A+( p&3) J, then w(#, {)>B(J) implies that # # 1 J, { . v For a multicurve 1 we give the vector space R 1 the sup norm and Hom(R 1, R 1 ) the corresponding norm, denoted by & } &. Propostion 2.4. Let k 1 be an integer and D=d Tf ({, _ f ({)). If #", #$ are nonperipheral simple closed curves in S 2 &P f such that some component of f &k (#$) is homotopic to #", then w(#", {) w(#$, {)&k(log d+2D).
(1)
Proof. Let X=P 1 &P be the Riemann surface underlying {, X$ be the Riemann surface underlying _ kf ({), and let f k{ be as in [1 Proposition 2.2] applied to f k. Then f k{ is analytic and defines a covering map from X"= P 1 &( f k{ ) &1 P Ä X. Since analytic coverings are isometries and the degree of f is d, l X" (#") d kl X (#$). The inclusion X" Ä X$ is length-decreasing, so l _ k({) (#") d kl { (#$). The bound now follows since _ f is distance-nonincreasf ing and the map { [ w(#, {) is Lipschitz with constant two [1, Proposition 7.2]. K 2.5. Reduction of Main Theorem The following is a refinement of [1, Proposition 8.2]; we use the same notation. The novel feature is that we allow J to vary. Proposition 2.5. There exists an integer m 1 1 depending only on d, p with the following property. Let D=d({, _ f ({)). For any J m 1 (log d+2D), there is a constant C(J) depending on J, p, d, D such that whenever w({)>C(J), 1=1 J, { {<, and if 1 {<, then w(1 , _ m1 ({)) w(1 , {). Proof. If w({) ( p&3) J+A=B(J) then the same arguments as in [1] show that 1=1 J, { is nonempty and is an f-stable multicurve. Suppose now that 1 {<. By Proposition 2.1 we have ( f 1 )=
\
f 1 C
0 f 12
+,
where & f 1 &<1. Since the set of possible f 1 's is finite, there is an integer m 1 depending only on p, d such that & f m 1 &<12 for all m m 1 . The remainder of the proof proceeds in exactly the same manner as in [1], using the crucial fact that the matrix for f 1 has the submatrix for f 1
CANONICAL THURSTON OBSTRUCTIONS
161
in the upper-left, so that the estimates [1, p. 288, top] may be applied to each coordinate corresponding to an element of 1 . Indeed, we may take
\
C(J)=sup log 2
\
2 pd m1 + , B(J) , ? L0
+
+
where L 0 =d m1e &B(J). K As a consequence, curves which become extraneous for some multicurve of the form 1 J, {i cannot, under iteration of _ f , become too short: Proposition 2.6 (Bound for Extraneous Curves). Let J m 1 (log d+2D). Suppose w({ 0 ) C(J ) + 2m 1 D. Let i 1 be the first index for which this occurs. By [1, Proposition 7.2] we have that w(1 , { i1 &m1 )>C(J). Then w(1 , { i1 ) w(1 , { i1 &m1 )C(J) B(J); cf. Subsection 2.4 and the previous proposition in which C(J) is defined. K We will derive our main theorem from the following proposition. Below, the constants ; and 2 are from Propositions 2.2 and 2.3, and \=0.548... is defined in the next section. Proposition 2.7 (Short Curves Remain Short). Let 1 be a multicurve for which 1=1 2 . Let { # Tf and let D=d Tf ({, _ f ({)). (1)
For all # # 1, the sequence [w(#, _ kf ({))] is bounded below.
(2) Suppose min # # 1 w(#, {) log(3;)+\. Write 1=1 $ ? 1 ", where 1$ is the union of the irreducible components 1 j of 1 for which *( f 1j ) 1. Then (a)
for all #$ # 1$, the sequence [w(#$, _ kf ({))] k 0
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KEVIN M. PILGRIM
is }-quasi-nondecreasing, where }=log ;&1&2\&2( p&3)(log d+2D). (b)
for all #" # 1 ", and all k 0, w(#", _ kf ({)) min [w(#$, _ kf ({))]&2(log d+2D). #$ # 1 $
(3)
Suppose min # # 1 w(#, {) J A , where J A =max[log(3;)+\, A]+}+2(log d+2D).
Then for all # # 1 and all k 0, we have w(#, _ kf ({)) A. Proof of Theorem 1.1. For any #, the map { [ w(#, {) is Lipschitz with constant two [2, Proposition 7.2]. Thus, the set 1 c is independent of the basepoint { 0 used in its definition. Moreover, if f is equivalent to a rational function, then _ f has a fixed point [1, Proposition 2.3] and so the lengths of geodesics remain bounded. Suppose now that f is not equivalent to a rational map. Choose { 0 with w({ 0 )E(J) for some n, so by Proposition 2.6 the set 1 J, {i , 2 of finite depth curves in 1 J, {i is nonempty for some i. Moreover, if # # 1 J, {i , 2 , then w(#, { i0 ) a+J>J A . Proposition 2.7(3) then implies 0 w(#, { i ) A for all i i 0 . It follows that 1 J =. 1 J, {i , 2 i
and G= . 1 J J JA
are multicurves, since # # 1 J satisfies w(#, { i ) A for all i sufficiently large. Since w(# n , { in ) Ä , given any fixed J J A , w(# n , { in ) E(J) for infinitely many n. Hence # n # 1 J /G infinitely often. Since G is finite, for some # # G we have # n =# infinitely often. Hence the set 1 u =[# | w(#, { i ) is unbounded] is nonempty.
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We claim that 1u = , 1J . J JA
The inclusion 1 u # J JA 1 J is clear. To see the other inclusion, let # # 1 u . Given J, choose i such that w(#, { i )>E(J). By Proposition 2.6, # # 1 J, {i , 2 , hence # # 1 J . We claim that in fact 1 u =1 Jc for some J c J A . Otherwise, for all J J A , there exists # J such that # J # 1 J /G but # J Â 1 u . Since G is finite, this implies that there is some # # G such that # # 1 J for arbitrarily large J, while also # Â 1 u . This is a contradiction, since # # 1 J for arbitrarily large J implies that the sequence [w(#, { i )] is unbounded. Hence there exists J c so that 1 u =1 Jc =. 1 Jc , {i , 2 . i
For each i for which 1=1 Jc , {i , 2 is nonempty, applying Proposition 2.7(2a), we see that if #$ # 1 $ then the sequence [w(#$, { i )] is both unbounded and quasi-nondecreasing, so by (2.3.2), w(#$, { i ) Ä . Conclusion (2b) of Proposition 2.7 then implies that w(#, { i ) Ä for all # # 1. Hence 1 u =1 c =[# | w(#, { i ) Ä ]. Finally, we claim that or some i=i c we have 1 c =1 Jc , {i , 2 . c
Since 1 c =1 u =. 1 Jc , {i , 2 i
the inclusion 1 c #1 Jc , {i , 2 holds for all i. Choose i c so large that for all # # 1 c and all i i c we have w(#, { ic )>E(J c ). By Proposition 2.6 we have # # 1 Jc , {i , 2 . Thus 1 c =1 Jc , ic , 2 , and so 1 c is a simple Thurston obstruction. c To prove the claim in Remark 1, suppose to the contrary that 1 is a simple obstruction and # c # 1 c intersects # # 1. Since l {i (# c ) Ä 0, we have by the Margulis lemma that l {i (#) Ä and so w(#, { i ) Ä & . This is impossible, by Proposition 2.7(1). K Proof of Theorem 1.2. If w(#, { i ) E(J c ) for some i then by Proposition 2.6 we have # # 1 Jc , {i , 2 . However, the proof of Theorem 1.1 showed
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that 1 c = i 1 Jc , {i , 2 , and so in this case we have # # 1 c . If w(#, { i )
3. PROOF OF PROPOSITION 2.7 Note that Proposition 2.7(2)(b) follows from Proposition 2.4 and 2.3. Given parts (2)(a) and (2)(b) of Proposition 2.7, part (3) follows immediately. Our bounds for parts (1) and (2)(a) will follow from estimates applied to moduli of certain annuli, rather than lengths of curves. The relationship between these is developed thoroughly in [1, Sect. 6], which we now summarize. Annuli on Riemann Surfaces. For { # Tf let (A(#), {) be the Riemann surface, conformally isomorphic to an annulus, obtained by taking the covering space of { corresponding to the Z-subgroup of the fundamental group of { generated by #. The core curve of (A(#), {) is a geodesic of length l { (#) and thus mod(A(#), {)=
? . 2l { (#)
(2)
If A 1, ..., A n are disjoint annuli on { with core curves homotopic to #, then mod(A 1 )+ } } } +mod(A n ) mod(A(#), {).
(3)
By [1, Proposition 6.1(c)], if # is a geodesic of hyperbolic length l on the Riemann surface defined {, then there is a canonically defined annulus which we denote by (a(#), {) (denoted A l in [1]) of modulus m(l ) which satisfies ? ? &1 m(l ) . 2l 2l
(4)
mod(A(#), {)&1
(5)
w(#, {) log(4?)=0.242...,
(V)
Thus, for all {, we have
If
CANONICAL THURSTON OBSTRUCTIONS
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|log mod(a(#), {)&w(#, {)| <\,
(6)
then
where \=1&log(?2)=0.548... is a universal constant. To see this, note that the hypothesis (V) and inequality (5) imply that 1 mod(A(#), {)&1
log mod(A(#), {)&1
mod(a(1), {) # R 1+
the vectors of moduli (mod(A(#), {)) # # 1 and (mod(a(#), {)) # # 1 , respectively. Set mod(A(1), {)=min[mod(A(#), {)] ##1
and define mod(a(1 ), {) similarly. Lemma 3.1. Let ; be the constant as in Proposition 2.2. Let 1 be an irreducible multicurve containing no extraneous curves. Then for all { and all k 0, (1)
mod(A(1 ), _ kf ({)) ; mod(a(1), {), and
(2)
mod(a(1), _ kf ({)) ; mod(a(1), {)&1.
Proof. Since thee are no extraneous curves, the leading eigenvalue of f 1 is at least one. Let v be the unique positive eigenvector of sup norm one corresponding to the leading eigenvalue of A=( f 1 ) and let 1 denote the vector all of whose coordinates are equal to one. By [1, Theorem 6.3], the Riemann surface for { contains pairwise disjoint annuli of modulus mod(a(#), {) about each element of 1. For vectors x, y, we say x y if x& y has nonnegative entries. Then mod(a(1 ), {) mod(a(1 ), {) 1 mod(a(1 ), {) v
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KEVIN M. PILGRIM
and so by applying (3) to preimages of these annuli under f k{ , mod(A(1 ), _ kf ({))>A k mod(a(1 ), {) A k mod(a(1 ), {) v mod(a(1), {) v ; mod(a(1 ), {) 1. Hence for all # # 1, mod(A(#), _ kf ({)) ; mod(a(1 ), {). The bound (5) implies the second conclusion.
K
Proof of Proposition 2.7, Part (1). Write 1=1 $ ? 1 " where 1 $ is the union of the irreducible components of 1 having leading eigenvalue at least one. Applying Lemma 3.1 to each such component, we find that for all k 0 mod(A(1 $), _ kf ({)) ; mod(a(1 $), {). For #$ # 1 $ this gives an upper bound on l _k { (#$), hence a lower bound on f w(#$, _ kf {). For #" # 1 ", a lower bound for w(#", _ kf {) follows from Proposition 2.7(2)(b). K Lemma 3.2.
If a, b>0, 0<; 1, and exp(a) ; exp(b)&1
then a&b log ;&1. Proof.
d log x 1 on [1, ) we have If ; exp(b)&1 1 then since 0< dx
log( ; exp(b)&1) log ;+b&1. Hence exp(a) ; exp(b)&1 O a=log exp(a) log(; exp(b)&1) log ;+ b&1 O a log ;+b&1 O a&b log ;&1. If ; exp(b)&1<1 then b0, a&b>0&b= &b>log ;&log 2>log ;&1. K Lemma 3.3. Let 1 be as in Lemma 3.1. Let w(1, {)=min # # 1 w(#, {). If w(1, {) log(3;)+\, then the sequence w(1, _ kf ({)) is (log ;&1&2\) quasi-nondecreasing.
CANONICAL THURSTON OBSTRUCTIONS
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Proof. Suppose w(1, {) log(3;)+\. Inequality (6) then implies that log mod(a(1), {) log(3;), hence that mod(a(1 ), {) 3;. So ; mod(a(1 ), {)&1 2. By Lemma 3.1, we then have for all k 0 that mod(a(1), _ kf ({)) 2.
(7)
Now consider the sequence [x k =log mod(a(1 ), _ kf {)]. Choose arbitrarily k 2 >k 1 0 and set a=x k2 , b=x k1 . By the previous paragraph a, b>0. Lemma 3.1, applied with k=k 2 &k 1 and {=_ kf 1 ({), implies that exp(a) ; exp(b)&1. Lemma 3.2 then implies that a&b log ;&1, so the sequence [x k ] is (log ;&1)-quasi-nondecreasing. By Eq. (7) and Inequality (5), mod(A(1), _ kf ({)) 2.
(8)
Using (2) and (8) we have log(?2)+w(1, _ kf ({)) log 2, and so w(1, _ kf ({)) that log(4?). The bound (6) then implies |x k &w(1, _ kf ({))| <\. By (2.3.1) the sequence [w(1, _ kf ({))] is thus (log ;&1&2\)-quasi-non decreasing. K Lemma 3.4.
If 1 is an irreducible multicurve, then for all # i , # j # 1, |w(# i , {)&w(# j , {)| ( p&3)(log d+2D).
Proof. Since 1 is irreducible, there is an integer q |1 | p&3 such that # i is homotopic to a component of f &q (# j ). Applying Inequality (1), we see that w(# i , {) w(# j , {) & ( p & 3)(log d + 2D). The bound now follows by symmetry. K Proof of Proposition 2.7, Part (2)(a). Suppose w(1, {) log(3;)+\. which *(1 ) 1. By Let [1 j ] be the set of irreducible components of 1 $ for j Lemma 3.3, for each 1 j , the sequence [w(1 j , _ kf ({))] is (log ;&1&2\)quasi-nondecreasing. By Lemma 3.4 and (2.3.1), for each # # 1 j , the sequence [w(#, _ kf ({))] is }=(log ;&1&2\&2( p&3)(log d+2D))-quasinondecreasing. K
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REFERENCES 1. A. Douady and J. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171, No. 2 (1993), 263297. 2. F. R. Gantmacher, ``Theory of Matrices,'' Chelsea, New York, 1959.
Canonical Thurston Obstructions Kevin M. Pilgrim 1 Department of Mathematics and Statistics, University of Missouri at Rolla, Rolla, Missouri 65409-0020 E-mail: pilgrimumr.edu Communicated by R. D. Mauldin Received October 19, 1999; accepted October 25, 2000
We refine Douady and Hubbard's proof of Thurston's topological characterization of rational functions by proving the following theorem. Let f: S 2 Ä S 2 be a branched covering with finite postcritical set P f and hyperbolic orbifold. Let 1 c denote the set of all homotopy classes # of nonperipheral, simple closed curves in S 2 &P f such that the length of the unique geodesic homotopic to # tends to zero under iteration of the Thurston map induced by f on Teichmuller space. Then either 1 c is empty, and f is equivalent to a rational function, or else 1 c is a Thurston obstruction. 2001 Academic Press
1. INTRODUCTION For relevant background we refer the reader to [1]. Let f: S 2 Ä S 2 be an orientation-preserving branched covering map of degree d 2, and let P f = n>0 f b n (0 f ) where 0 f is the set of critical points of f. We say f is a Thurston map if P f is finite; we assume this throughout this work. Two Thurston maps f, g are combinatorially equivalent if there are homeomorphisms %, %$: (S 2, P f ) Ä (S 2, P g ) such that % is isotopic to %$ relative to P f and % b f =g b %$. The orbifold Of associated to f is the topological orbifold with underlying space S 2 and whose weight &(x) at x is the least common multiple of the local degree of f over all iterated preimages of x. The orbifold Of is said to be hyperbolic if the Euler characteristic /(Of )=2& x # Pf (1&1&(x)) is negative. A simple closed curve # # S 2 &P f is nonperipheral if each component of 2 S &# contains at least two points of P f . A nonempty set 1=[# 1 , ..., # n ] of simple, closed, nonperipheral, non-homotopic curves on S 2 &P f we call a multicurve; it is said to be f-stable if for any # # 1, all components of 1
Research partially supported by NSF Grant DMS 9996070.
154 0001-870801 35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.
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f &1 (#) are either peripheral, or homotopic to an element of 1. Associated to an f-stable multicurve 1 is a linear map f1 : R 1 Ä R 1 defined as follows. Let # i, j, : be the components of f &1 (# j ) which are homotopic to # i in S 2 &P f . Define 1 #i , d i, : i, j, :
f 1 (# j )=:
where d i, j, : is the (positive) degree of the map f | # i, j, : : # i, j, : Ä # j . Then f 1 has spectral radius realized by a real nonnegative eigenvalue *( f, 1 ). Thurston's characterization is Theorem (Thurston's Characterization and Rigidity Theorem). A Thurston map f with hyperbolic orbifold is equivalent to a rational function if and only if for any f-stable multicurve 1 we have *( f, 1 )<1. In that case, the rational function is unique up to conjugation by an automorphism of the Riemann sphere. If Of is hyperbolic and 1 is an f-stable multicurve with *( f, 1 ) 1 we will call 1 a Thurston obstruction. This is something of a misnomer; in Section 2 we introduce the idea of a simple Thurston obstruction which is better suited for our purposes. The idea of the proof is the following. Associated to f is a Teichmuller space Tf modelled on (S 2, P f ), and an analytic self-map _ f : Tf Ä Tf . The existence of a rational map combinatorially equivalent to f is equivalent to the existence of a fixed point of _ f . The map _ f is distance-nonincreasing for the Teichmuller metric, and if the associated orbifold is hyperbolic, _ 2f decreases distances, though not necessarily uniformly. To find a fixed point, one chooses arbitrarily { 0 # Tf and considers the sequence { i =_ fb i ({ 0 ). If [{ i ] fails to converge, then the length of the shortest geodesic on { i , in its natural hyperbolic metric, must become arbitrarily small. In this case, for some i sufficiently large, the family of geodesics on { i which are both sufficiently short and sufficiently shorter than any other geodesics on { i form a Thurston obstruction. For a nonperipheral simple closed curve #/S 2 &P f let l { (#) denote the hyperbolic length of the unique geodesic on the marked Riemann surface given by { which is homotopic to #. In [1] the authors show by example that it is possible for curves of two different obstructions to intersect, thus preventing their lengths from becoming simultaneously small. Hence, if 1 is an obstruction and # # 1, then it is not necessarily true that inf i [l {i (#)]= 0. Moreover, their proof does not explicitly show that if f is obstructed,
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KEVIN M. PILGRIM
then inf i [l {i (#)]=0 for some fixed curve #. Thus it is conceivable that, for each i, there is a curve # i such that inf[l {i (# i )]=0 i
while for fixed i inf[l {j (# i )]>0. j
We will rule out this possibility, and in so doing establish Theorem 1.1 (Canonical Obstruction). Let f be a Thurston map with hyperbolic orbifold, and let 1 c denote the set of all homotopy classes of nonperipheral, simple closed curves # in S 2 &P f such that l {i (#) Ä 0 as i Ä . (1)
If 1 c is empty, then f is combinatorially equivalent to a rational
map. (2) Otherwise, 1 c is an f-stable multicurve for which *( f, 1 c ) 1, and hence is a canonically defined Thurston obstruction to the existence of a rational map combinatorially equivalent to f. The proof also shows, with the same hypotheses, Theorem 1.2 (Curves Degenerate or Stay Bounded). Let # be a nonperipheral, simple closed curve in S 2 &P f . (1)
If # # 1 c , then l {i (#) Ä 0 as i Ä .
(2) If # Â 1 c , then l {i (#) E for all i, where E is a positive constant depending on { 0 but not on #. Remarks. (1) In fact, 1 c has the additional property of being simple (see Subsection 2.2) and is also disjoint from all other simple obstructions. The idea of non-simple is the following. To an obstruction, add a disjoint invariant multicurve which is not an obstruction. The added curves are in a sense extraneous. A simple obstruction contains no such extraneous curves. (2) Our arguments show that 1 c equals the set of curves which become arbitrarily short. (3) It is tempting to view the obstruction 1 c as the analog, for branched coverings, of the tori in the canonical torus decomposition for irreducible three-manifolds; this is the subject of work in progress. (4) We do not know if _ f extends in a reasonable way to one of the natural compactifications of Tf .
CANONICAL THURSTON OBSTRUCTIONS
157
(5) Our proof uses no analytic tools other than those in [1] and is based mostly on combinatorial refinements of their arguments. Outline of This Paper. In Section 2 we introduce our combinatorial refinements and reduce the proof of Theorem 1.1 to Proposition 2.7, Short Curves Remain Short, which we prove in Section 3.
2. BACKGROUND 2.1. Nonnegative Matrices We shall need some basic facts from the theory of nonnegative square matrices; see [2, Vol. 2, Chap. XIII]. Such a matrix A always has a largest nonnegative (or leading) eigenvalue *(A) equal to its spectral radius. Irreducible Matrices. A nonnegative n-by-n matrix A is called irreducible if no permutation of the indices places it in block lower-triangular form. In this case, given i, j there exists q with 0 q n such that (Aq ) i, j {0. By the Frobenius Theorem, A has a positive eigenvalue *(A) equal to its spectral radius, and up to scale, there is exactly one positive eigenvector corresponding to *(A). Reducible Matrices. By applying a permutation of the indices, A may be assumed to be in the form
\
A 11 A 21 }}} As1
0 A 22 }}} As2
}}} }}} }}} }}}
0 0 , 0 A ss
+
where the blocks A jj are irreducible. Moreover, *(A)=sup *(A jj ). j
2.2. Multicurves We shall need to consider transformations for multicurves 1 which are not necessarily f-stable. Define f 1 : R 1 Ä R 1 by the same formula. We say 1 is irreducible if the matrix ( f 1 ) is irreducible. Let 1 be a multicurve and A=( f 1 ). Let 1 j denote the set of indices (curves in 1) corresponding to the jth block in the decomposition given in the previous subsection, so that A jj =( f 1j ). We call the 1 j 's the irreducible components of 1. As sets, they are well-defined up to permutations of the indices j.
158 Depth. set
KEVIN M. PILGRIM
Let 1 be a multicurve, A=( f 1 ), and suppose *(A) 1. Then the
1 $= .
1j
*(Ajj ) 1
is nonempty. Definition 2.1 (Depth, Simple Obstruction). Let # # 1. The depth of # with respect to 1 is the smallest integer k 0 for which there exists #$ # 1 $ with (A k ) ##$ {0. If no such k exists, we say # has infinite depth and call # extraneous with respect to 1. The set of elements of 1 of finite and infinite depth we denote by 1 2 and 1 , respectively. A Thurston obstruction 1 is called simple if 1=1 2 . Thus, the depth is how many backwards iterates are necessary to make # equal to the preimage of some curve #$ # 1 $. In particular, curves of depth zero lie in 1 $. Note that the depth of a curve depends on the ambient multicurve and may change if curves are added or subtracted. Proposition 2.1. If 1 is a Thurston obstruction, then 1 2 is a simple Thurston obstruction. In particular, after applying a permutation, ( f 1 )=
f 1
\C
0 , f 12
+
where *( f 1 )<1 and *( f 12 )=*( f 1 ) 1. Proof. A preimage of a curve of finite depth is of finite depth, so 1 2 is f-stable and the block-lower-triangular form holds. Furthermore, the set [1 j ] of irreducible components of 1 is the disjoint union of the set of irreducible components for 1 and 1 2 . So *( f 1j )<1 for all 1 j /1 and *( f 1j )=*( f 1 ) for some 1 j /1 2 . Hence *( f 1 )<1 and *( f 12 )=*( f 1 ). K Finiteness Results. A generalization on [1, Lemma 1.2] is that the set of possible transformations f 1 depends only on the degree d=deg f and the cardinality p of P f . As consequences, we have Proposition 2.2. There is a number ; depending only on d, p with 0< ; 1 such that if 1 is any irreducible multicurve for which *( f 1 ) 1, and if v is the unique positive eigenvector of f 1 with sup norm equal to 1, then the smallest coordinate of v is bounded below by ;.
CANONICAL THURSTON OBSTRUCTIONS
159
Proposition 2.3. There is a positive integer 2 depending only on p, d such that whenever # has finite depth with respect to some 1, then its depth is less than or equal to 2. In fact, 2 p&4. 2.3. Quasi-Nondecreasing Sequences Definition 2.2. Let } # R. A sequence [x i ] i=0 of real numbers is called }-quasi-nondecreasing if for all i 1
If [x i ] is quasi-nondecreasing and unbounded then x i Ä + .
2.4. Notation and Basic Facts Throughout, we will use the notation and basic results of [1]. In particular: v p=*P f , d=deg f. v {, an element of Tf . v w(#, {)=&log l { (#). v L { =[w(#, {)]/R, where # ranges over all nonperipheral simple closed curves in S 2 &P f . v w({)=sup #[w(#, {)]. v w(1, {)=sup # # 1 [w(#, {)]. v A=&log log(- 2+1)=0.126..., the smallest value of w(#, {) guaranteeing that the geodesic homotopic to # on { is simple and disjoint from any other geodesic #$ with w(#$, {) A. Thus for any { there are at most p&3 curves # with w(#, {) A. v For J>0 let ]a, b[ be the lowest interval in R&L { of length J, where a A, and define 1 J, { =[# | w(#, {) b].
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KEVIN M. PILGRIM
Thus, 1 J, { is the multicurve consisting of the set of all geodesics on { which are both sufficiently short and sufficiently shorter than any other curves. Note that if B(J)=A+( p&3) J, then w(#, {)>B(J) implies that # # 1 J, { . v For a multicurve 1 we give the vector space R 1 the sup norm and Hom(R 1, R 1 ) the corresponding norm, denoted by & } &. Propostion 2.4. Let k 1 be an integer and D=d Tf ({, _ f ({)). If #", #$ are nonperipheral simple closed curves in S 2 &P f such that some component of f &k (#$) is homotopic to #", then w(#", {) w(#$, {)&k(log d+2D).
(1)
Proof. Let X=P 1 &P be the Riemann surface underlying {, X$ be the Riemann surface underlying _ kf ({), and let f k{ be as in [1 Proposition 2.2] applied to f k. Then f k{ is analytic and defines a covering map from X"= P 1 &( f k{ ) &1 P Ä X. Since analytic coverings are isometries and the degree of f is d, l X" (#") d kl X (#$). The inclusion X" Ä X$ is length-decreasing, so l _ k({) (#") d kl { (#$). The bound now follows since _ f is distance-nonincreasf ing and the map { [ w(#, {) is Lipschitz with constant two [1, Proposition 7.2]. K 2.5. Reduction of Main Theorem The following is a refinement of [1, Proposition 8.2]; we use the same notation. The novel feature is that we allow J to vary. Proposition 2.5. There exists an integer m 1 1 depending only on d, p with the following property. Let D=d({, _ f ({)). For any J m 1 (log d+2D), there is a constant C(J) depending on J, p, d, D such that whenever w({)>C(J), 1=1 J, { {<, and if 1 {<, then w(1 , _ m1 ({)) w(1 , {). Proof. If w({) ( p&3) J+A=B(J) then the same arguments as in [1] show that 1=1 J, { is nonempty and is an f-stable multicurve. Suppose now that 1 {<. By Proposition 2.1 we have ( f 1 )=
\
f 1 C
0 f 12
+,
where & f 1 &<1. Since the set of possible f 1 's is finite, there is an integer m 1 depending only on p, d such that & f m 1 &<12 for all m m 1 . The remainder of the proof proceeds in exactly the same manner as in [1], using the crucial fact that the matrix for f 1 has the submatrix for f 1
CANONICAL THURSTON OBSTRUCTIONS
161
in the upper-left, so that the estimates [1, p. 288, top] may be applied to each coordinate corresponding to an element of 1 . Indeed, we may take
\
C(J)=sup log 2
\
2 pd m1 + , B(J) , ? L0
+
+
where L 0 =d m1e &B(J). K As a consequence, curves which become extraneous for some multicurve of the form 1 J, {i cannot, under iteration of _ f , become too short: Proposition 2.6 (Bound for Extraneous Curves). Let J m 1 (log d+2D). Suppose w({ 0 )
For all # # 1, the sequence [w(#, _ kf ({))] is bounded below.
(2) Suppose min # # 1 w(#, {) log(3;)+\. Write 1=1 $ ? 1 ", where 1$ is the union of the irreducible components 1 j of 1 for which *( f 1j ) 1. Then (a)
for all #$ # 1$, the sequence [w(#$, _ kf ({))] k 0
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KEVIN M. PILGRIM
is }-quasi-nondecreasing, where }=log ;&1&2\&2( p&3)(log d+2D). (b)
for all #" # 1 ", and all k 0, w(#", _ kf ({)) min [w(#$, _ kf ({))]&2(log d+2D). #$ # 1 $
(3)
Suppose min # # 1 w(#, {) J A , where J A =max[log(3;)+\, A]+}+2(log d+2D).
Then for all # # 1 and all k 0, we have w(#, _ kf ({)) A. Proof of Theorem 1.1. For any #, the map { [ w(#, {) is Lipschitz with constant two [2, Proposition 7.2]. Thus, the set 1 c is independent of the basepoint { 0 used in its definition. Moreover, if f is equivalent to a rational function, then _ f has a fixed point [1, Proposition 2.3] and so the lengths of geodesics remain bounded. Suppose now that f is not equivalent to a rational map. Choose { 0 with w({ 0 )
and G= . 1 J J JA
are multicurves, since # # 1 J satisfies w(#, { i ) A for all i sufficiently large. Since w(# n , { in ) Ä , given any fixed J J A , w(# n , { in ) E(J) for infinitely many n. Hence # n # 1 J /G infinitely often. Since G is finite, for some # # G we have # n =# infinitely often. Hence the set 1 u =[# | w(#, { i ) is unbounded] is nonempty.
CANONICAL THURSTON OBSTRUCTIONS
163
We claim that 1u = , 1J . J JA
The inclusion 1 u # J JA 1 J is clear. To see the other inclusion, let # # 1 u . Given J, choose i such that w(#, { i )>E(J). By Proposition 2.6, # # 1 J, {i , 2 , hence # # 1 J . We claim that in fact 1 u =1 Jc for some J c J A . Otherwise, for all J J A , there exists # J such that # J # 1 J /G but # J Â 1 u . Since G is finite, this implies that there is some # # G such that # # 1 J for arbitrarily large J, while also # Â 1 u . This is a contradiction, since # # 1 J for arbitrarily large J implies that the sequence [w(#, { i )] is unbounded. Hence there exists J c so that 1 u =1 Jc =. 1 Jc , {i , 2 . i
For each i for which 1=1 Jc , {i , 2 is nonempty, applying Proposition 2.7(2a), we see that if #$ # 1 $ then the sequence [w(#$, { i )] is both unbounded and quasi-nondecreasing, so by (2.3.2), w(#$, { i ) Ä . Conclusion (2b) of Proposition 2.7 then implies that w(#, { i ) Ä for all # # 1. Hence 1 u =1 c =[# | w(#, { i ) Ä ]. Finally, we claim that or some i=i c we have 1 c =1 Jc , {i , 2 . c
Since 1 c =1 u =. 1 Jc , {i , 2 i
the inclusion 1 c #1 Jc , {i , 2 holds for all i. Choose i c so large that for all # # 1 c and all i i c we have w(#, { ic )>E(J c ). By Proposition 2.6 we have # # 1 Jc , {i , 2 . Thus 1 c =1 Jc , ic , 2 , and so 1 c is a simple Thurston obstruction. c To prove the claim in Remark 1, suppose to the contrary that 1 is a simple obstruction and # c # 1 c intersects # # 1. Since l {i (# c ) Ä 0, we have by the Margulis lemma that l {i (#) Ä and so w(#, { i ) Ä & . This is impossible, by Proposition 2.7(1). K Proof of Theorem 1.2. If w(#, { i ) E(J c ) for some i then by Proposition 2.6 we have # # 1 Jc , {i , 2 . However, the proof of Theorem 1.1 showed
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KEVIN M. PILGRIM
that 1 c = i 1 Jc , {i , 2 , and so in this case we have # # 1 c . If w(#, { i )
3. PROOF OF PROPOSITION 2.7 Note that Proposition 2.7(2)(b) follows from Proposition 2.4 and 2.3. Given parts (2)(a) and (2)(b) of Proposition 2.7, part (3) follows immediately. Our bounds for parts (1) and (2)(a) will follow from estimates applied to moduli of certain annuli, rather than lengths of curves. The relationship between these is developed thoroughly in [1, Sect. 6], which we now summarize. Annuli on Riemann Surfaces. For { # Tf let (A(#), {) be the Riemann surface, conformally isomorphic to an annulus, obtained by taking the covering space of { corresponding to the Z-subgroup of the fundamental group of { generated by #. The core curve of (A(#), {) is a geodesic of length l { (#) and thus mod(A(#), {)=
? . 2l { (#)
(2)
If A 1, ..., A n are disjoint annuli on { with core curves homotopic to #, then mod(A 1 )+ } } } +mod(A n ) mod(A(#), {).
(3)
By [1, Proposition 6.1(c)], if # is a geodesic of hyperbolic length l on the Riemann surface defined {, then there is a canonically defined annulus which we denote by (a(#), {) (denoted A l in [1]) of modulus m(l ) which satisfies ? ? &1 m(l ) . 2l 2l
(4)
mod(A(#), {)&1
(5)
w(#, {) log(4?)=0.242...,
(V)
Thus, for all {, we have
If
CANONICAL THURSTON OBSTRUCTIONS
165
|log mod(a(#), {)&w(#, {)| <\,
(6)
then
where \=1&log(?2)=0.548... is a universal constant. To see this, note that the hypothesis (V) and inequality (5) imply that 1 mod(A(#), {)&1
log mod(A(#), {)&1
mod(a(1), {) # R 1+
the vectors of moduli (mod(A(#), {)) # # 1 and (mod(a(#), {)) # # 1 , respectively. Set mod(A(1), {)=min[mod(A(#), {)] ##1
and define mod(a(1 ), {) similarly. Lemma 3.1. Let ; be the constant as in Proposition 2.2. Let 1 be an irreducible multicurve containing no extraneous curves. Then for all { and all k 0, (1)
mod(A(1 ), _ kf ({)) ; mod(a(1), {), and
(2)
mod(a(1), _ kf ({)) ; mod(a(1), {)&1.
Proof. Since thee are no extraneous curves, the leading eigenvalue of f 1 is at least one. Let v be the unique positive eigenvector of sup norm one corresponding to the leading eigenvalue of A=( f 1 ) and let 1 denote the vector all of whose coordinates are equal to one. By [1, Theorem 6.3], the Riemann surface for { contains pairwise disjoint annuli of modulus mod(a(#), {) about each element of 1. For vectors x, y, we say x y if x& y has nonnegative entries. Then mod(a(1 ), {) mod(a(1 ), {) 1 mod(a(1 ), {) v
166
KEVIN M. PILGRIM
and so by applying (3) to preimages of these annuli under f k{ , mod(A(1 ), _ kf ({))>A k mod(a(1 ), {) A k mod(a(1 ), {) v mod(a(1), {) v ; mod(a(1 ), {) 1. Hence for all # # 1, mod(A(#), _ kf ({)) ; mod(a(1 ), {). The bound (5) implies the second conclusion.
K
Proof of Proposition 2.7, Part (1). Write 1=1 $ ? 1 " where 1 $ is the union of the irreducible components of 1 having leading eigenvalue at least one. Applying Lemma 3.1 to each such component, we find that for all k 0 mod(A(1 $), _ kf ({)) ; mod(a(1 $), {). For #$ # 1 $ this gives an upper bound on l _k { (#$), hence a lower bound on f w(#$, _ kf {). For #" # 1 ", a lower bound for w(#", _ kf {) follows from Proposition 2.7(2)(b). K Lemma 3.2.
If a, b>0, 0<; 1, and exp(a) ; exp(b)&1
then a&b log ;&1. Proof.
d log x 1 on [1, ) we have If ; exp(b)&1 1 then since 0< dx
log( ; exp(b)&1) log ;+b&1. Hence exp(a) ; exp(b)&1 O a=log exp(a) log(; exp(b)&1) log ;+ b&1 O a log ;+b&1 O a&b log ;&1. If ; exp(b)&1<1 then b
CANONICAL THURSTON OBSTRUCTIONS
167
Proof. Suppose w(1, {) log(3;)+\. Inequality (6) then implies that log mod(a(1), {) log(3;), hence that mod(a(1 ), {) 3;. So ; mod(a(1 ), {)&1 2. By Lemma 3.1, we then have for all k 0 that mod(a(1), _ kf ({)) 2.
(7)
Now consider the sequence [x k =log mod(a(1 ), _ kf {)]. Choose arbitrarily k 2 >k 1 0 and set a=x k2 , b=x k1 . By the previous paragraph a, b>0. Lemma 3.1, applied with k=k 2 &k 1 and {=_ kf 1 ({), implies that exp(a) ; exp(b)&1. Lemma 3.2 then implies that a&b log ;&1, so the sequence [x k ] is (log ;&1)-quasi-nondecreasing. By Eq. (7) and Inequality (5), mod(A(1), _ kf ({)) 2.
(8)
Using (2) and (8) we have log(?2)+w(1, _ kf ({)) log 2, and so w(1, _ kf ({)) that log(4?). The bound (6) then implies |x k &w(1, _ kf ({))| <\. By (2.3.1) the sequence [w(1, _ kf ({))] is thus (log ;&1&2\)-quasi-non decreasing. K Lemma 3.4.
If 1 is an irreducible multicurve, then for all # i , # j # 1, |w(# i , {)&w(# j , {)| ( p&3)(log d+2D).
Proof. Since 1 is irreducible, there is an integer q |1 | p&3 such that # i is homotopic to a component of f &q (# j ). Applying Inequality (1), we see that w(# i , {) w(# j , {) & ( p & 3)(log d + 2D). The bound now follows by symmetry. K Proof of Proposition 2.7, Part (2)(a). Suppose w(1, {) log(3;)+\. which *(1 ) 1. By Let [1 j ] be the set of irreducible components of 1 $ for j Lemma 3.3, for each 1 j , the sequence [w(1 j , _ kf ({))] is (log ;&1&2\)quasi-nondecreasing. By Lemma 3.4 and (2.3.1), for each # # 1 j , the sequence [w(#, _ kf ({))] is }=(log ;&1&2\&2( p&3)(log d+2D))-quasinondecreasing. K
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REFERENCES 1. A. Douady and J. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171, No. 2 (1993), 263297. 2. F. R. Gantmacher, ``Theory of Matrices,'' Chelsea, New York, 1959.