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BODY %!PS-Adobe-2.0 %%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software %%Title: treeB.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o treeB.ps treeB %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1998.06.16:1021 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N 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Fv(1])18 b Fr(!)1255 1774 y Fn(j)1245 1784 y Fv(C)i(b)q(e)f(a)h(w)o(eigh)o(t.)29 b(W)l(e)19 b(denote)h(b)o(y)e Ft(f)1942 1766 y Fm(\016)p Fo(n)-108 1844 y Fv(the)f(map)g Ft(f)g Fr(\016)11 b Ft(f)17 b Fr(\016)12 b Ft(:)c(:)g(:)j Fr(\016)g Ft(f)23 b Fv(comp)q(osed)16 b Ft(n)i Fv(times.)k(If)16 b(eac)o(h)h(iterate)f(of)i Ft(f)k Fv(has)c(\014nitely)e(man)o(y)f(\014xed)i(p)q(oin)o(ts,)g(it)g (is)-108 1904 y(natural)g(to)g(de\014ne)f(the)g Fq(weighte)n(d)i(R)o (uel)r(le)h(zeta)f(function)g Fv(:)522 2054 y Ft(\020)543 2061 y Fo(f)r(;g)592 2054 y Fv(\()p Ft(t)p Fv(\))13 b(=)h(exp)796 2007 y Fl(X)799 2112 y Fo(n)p Fm(\025)p Fp(1)881 2020 y Ft(t)899 2002 y Fo(n)p 881 2043 42 2 v 887 2088 a Ft(n)977 2007 y Fl(X)936 2113 y Fo(x)p Fp(:)p Fo(f)987 2103 y Fn(\016)p Fk(n)1023 2113 y Fo(x)p Fp(=)p Fo(x)1098 1992 y(n)p Fm(\000)p Fp(1)1100 2007 y Fl(Y)1099 2113 y Fo(k)q Fp(=0)1173 2054 y Ft(g)r Fv(\()p Ft(f)1246 2033 y Fm(\016)p Fo(k)1286 2054 y Ft(x)p Fv(\))8 b Ft(:)-108 2210 y Fv(If)16 b Ft(g)g 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(Mathieu.Baillif@m)o(ath.uni)o(ge.c)o(h)926 2819 y Fv(1)p eop %%Page: 2 2 2 1 bop -108 286 a Fv(and)17 b(analogously)h([5],)d([6].)21 b(In)16 b(the)g(early)f(sev)o(en)o(ties,)f(Milnor)i(and)g(Th)o(urston)h (w)o(ere)f(able)g(to)g(relate)g(a)h(reduced)-108 347 y(un)o(w)o(eigh)o(ted)h(zeta)h(function)g Ft(\020)472 354 y Fo(f)495 347 y Fv(\()p Ft(t)p Fv(\))f(\(in)h(the)g(case)g(where)f Ft(f)25 b Fv(is)19 b(con)o(tin)o(uous)g(in)f([0)p Ft(;)8 b Fv(1])19 b(and)h(strictly)d(monotone)-108 407 y(on)g(\014nitely)d (man)o(y)h(subin)o(terv)m(als)g(of)h([0)p Ft(;)8 b Fv(1]\))16 b(to)g(the)f(determinan)o(t)f(\001)1170 414 y Fo(f)1192 407 y Fv(\()p Ft(t)p Fv(\))i(of)g(a)g(\014nite)f Fq(kne)n(ading)j (matrix)33 b Fv(whose)-108 467 y(co)q(e\016cien)o(ts)17 b(are)h(p)q(o)o(w)o(er)g(series)f(in)g Ft(t)p Fv(.)26 b(In)17 b(their)g(reduced)g(zeta)h(function,)f(Milnor)g(and)i(Th)o (urston)f(considered)-108 527 y(only)h(the)f(\014xed)f(p)q(oin)o(ts)i (of)g(negativ)o(e)e(t)o(yp)q(e,)h(i.e.)26 b(the)18 b(\014xed)g(p)q(oin) o(ts)h Ft(x)f Fv(where)f Ft(f)1396 509 y Fm(\016)p Fo(n)1456 527 y Fv(is)h(decreasing)g(throughout)-108 587 y(a)k(neigh)o(b)q(orho)q (o)q(d)h(of)e Ft(x)p Fv(,)h(but)f(they)g(\\coun)o(ted")g(them)f(t)o(w)o (o)h(times,)f(\\since)h(b)q(et)o(w)o(een)f(an)o(y)h(t)o(w)o(o)g (\014xed)f(p)q(oin)o(ts)-108 648 y(of)f(negativ)o(e)e(t)o(yp)q(e)h (there)f(m)o(ust)g(b)q(e)i(at)f(least)g(one)g(other)h(\014xed)e(p)q (oin)o(t.")28 b(The)18 b(p)q(o)o(w)o(er)g(series)g(are)g(obtained)g(b)o (y)-108 708 y(lo)q(oking)h(at)f(the)g(orbits)h(of)f(the)g(righ)o(thand) g(and)h(lefthand)f(limits)d(of)k(the)f(turning)g(p)q(oin)o(ts)g(\(the)g (endp)q(oin)o(ts)h(of)-108 768 y(the)f(subin)o(terv)m(als)g(cited)f(ab) q(o)o(v)o(e\))h(under)h(the)e(iterates)h(of)h Ft(f)5 b Fv(.)27 b(The)18 b(relation)g(b)q(et)o(w)o(een)f Ft(\020)1553 775 y Fo(f)1594 768 y Fv(and)i(\001)1732 775 y Fo(f)1755 768 y Fv(,)f(whic)o(h)f(w)o(e)-108 828 y(will)f(call)f Fq(the)j(Milnor-Thurston)g(r)n(elation)e Fv(is)714 938 y Ft(\020)735 945 y Fo(f)758 938 y Fv(\()p Ft(t)p Fv(\)\001)855 945 y Fo(f)877 938 y Fv(\()p Ft(t)p Fv(\))d(=)h(1)d(+)g Ft(p)p Fv(\()p Ft(t)p Fv(\))523 b(\(1\))-108 1048 y(with)17 b Ft(p)p Fv(\()p Ft(t)p Fv(\))f(a)g(p)q(olynomial)f(correcting)h (factor)h(\(See)e([11)q(]\).)-108 1108 y(Baladi)20 b(and)g(Ruelle)e ([5])h(later)g(considered)h Ft(f)25 b Fv(piecewise)18 b(con)o(tin)o(uous)h(functions)h(strictly)e(monotone)i(in)f(the)-108 1169 y(subin)o(terv)m(als)14 b(de\014ned)h(b)o(y)e(the)i(turning)f(p)q (oin)o(ts,)h(extended)e(the)h(Milnor-Th)o(urston)h(kneading)f(matrix)f (and)i(the)-108 1229 y(reduced)h(zeta)h(function)f(to)h(a)g(w)o(eigh)o (ted)f(case,)g(and)h(they)f(sho)o(w)o(ed)h(that)g(the)f(Milnor-Th)o (urston)h(relation)f(w)o(as)-108 1289 y(still)d(v)m(alid,)h Fq(under)i(the)g(assumption)f(that)h(the)g(weight)h Ft(g)g Fq(b)n(e)f(c)n(onstant)g(on)g(e)n(ach)g(monotonicity)g(interval)f Fv(|)f(i.e.,)-108 1349 y Ft(g)19 b Fv(lo)q(cally)c(constan)o(t.)-108 1409 y(Baladi)21 b([2])f(then)h(sho)o(w)o(ed)g(that)g(\(1\))g(still)f (holds)h(when)g Ft(g)i Fv(is)e(of)g(b)q(ounded)h(v)m(ariation)f(\(plus) g(some)f(tec)o(hnical)-108 1469 y(assumptions\),)15 b(b)o(y)e (considering)i(in\014nite)e(\(coun)o(table\))h(matrices.)19 b(More)14 b(precisely)l(,)e(she)i(built)g(\014nite)g(matrices)-108 1530 y(with)i(determinan)o(t)d(\001)318 1541 y Fo(f)r(;g)365 1531 y Fg(\()p Fk(n)p Fg(\))426 1530 y Fv(and)j(zeta)f(functions)g Ft(\020)852 1541 y Fo(f)r(;g)899 1531 y Fg(\()p Fk(n)p Fg(\))962 1530 y Fv(corresp)q(onding)h(to)g(increasingly)e(b)q(etter)h (appro)o(xima-)-108 1595 y(tions)d Ft(g)32 1577 y Fp(\()p Fo(n)p Fp(\))94 1595 y Fv(of)f Ft(g)i Fv(and)f(sho)o(w)o(ed)f(that)g (these)g(t)o(w)o(o)g(sequences)f(con)o(v)o(erge)g(as)h(analytic)f(|)h (resp)q(ectiv)o(ely)e(meromorphic)-108 1655 y(|)16 b(functions)h(in)f (a)g(disk,)g(to)g(\001)481 1662 y Fo(f)r(;g)546 1655 y Fv(and)h Ft(\020)662 1662 y Fo(f)r(;g)727 1655 y Fv(for)f(whic)o(h)g (the)g(form)o(ula)f(is)h(th)o(us)g(v)o(eri\014ed.)-108 1715 y(F)l(or)e(further)g(results,)g(see)f(also)i([4],)e([6],)g([12)q (].)20 b(The)14 b(aim)e(of)i(this)g(presen)o(t)f(pap)q(er)i(is)e(to)i (extend)e(the)h(results)f(of)h([5])-108 1775 y(and)h([2)o(])f(to)f (functions)h Ft(f)19 b Fv(:)13 b Ft(T)21 b Fr(!)13 b Ft(T)7 b Fv(,)13 b(where)g Ft(T)20 b Fv(is)14 b(a)g(ro)q(oted)g(orien)o (ted)e(tree)h(with)h(\014nitely)e(man)o(y)g(edges.)20 b(Sections)-108 1835 y(2)d(and)g(3)f(are)g(dev)o(oted)f(to)i(the)f (case)g(of)g Ft(g)i Fv(b)q(eing)f(piecewise)d(constan)o(t.)22 b(T)l(o)16 b(de\014ne)g(the)g(zeta)g(function)f(and)i(the)-108 1896 y(kneading)d(matrix)e(of)i Ft(f)5 b Fv(,)14 b(w)o(e)f(de\014ne)g (the)h Fq(addr)n(ess)e Fv(of)i(a)g(p)q(oin)o(t)g Ft(x)f Fv(in)h(the)f(tree)g Ft(T)7 b Fv(,)13 b(follo)o(wing)g(the)h(strategy)g (of)f([11)q(].)-108 1956 y(W)l(e)h(giv)o(e)f(here)g(t)o(w)o(o)h(p)q (ossible)g(de\014nitions)g(\(there)f(are)h(certainly)e(others\),)i (whic)o(h)g(eac)o(h)f(ha)o(v)o(e)g(adv)m(an)o(tadges)i(and)-108 2016 y(disadv)m(an)o(tages)i(:)j(the)15 b(\014rst)h(de\014nition)e(is)h (quite)f(elegan)o(t)h(and)g(natural)h(but)f(leads)g(to)h(a)f (correcting)f(factor)i Ft(p)p Fv(\()p Ft(t)p Fv(\))-108 2076 y(in)j(the)f(form)o(ula)f(\(1\).)28 b(This)18 b(factor)h(is)f(not) h(presen)o(t)f(when)g(w)o(e)g(tak)o(e)g(our)h(second)g(de\014nition)f (of)g(the)h(address,)-108 2136 y(whic)o(h)e(is)h(ho)o(w)o(ev)o(er)e (more)g(arti\014cial.)25 b(In)17 b(the)g(case)h(where)f Ft(T)24 b Fv(is)17 b(homeomorphic)f(to)i(the)f(in)o(terv)m(al,)f(b)q (oth)j(de\014-)-108 2197 y(nitions)e(reduce)e(to)i(the)f(address,)g (and)h(th)o(us)f(the)g(determinan)o(t)e(of)j([2])f([5].)-108 2257 y(Our)j(main)d(result)i(is)g(the)g(Milnor-Th)o(urston)g(relation)g (\(1\))g(for)h(a)f(map)g Ft(f)23 b Fv(of)c(a)f(ro)q(oted,)h(orien)o (ted)e(tree)h(and)g(a)-108 2317 y(lo)q(cally)e(constan)o(t)h(w)o(eigh)o (t)e Ft(g)r Fv(.)-108 2377 y(In)h(Section)g(4)h(w)o(e)f(brie\015y)f (describ)q(e)h(ho)o(w)g(to)h(adapt)g([2])f(to)g(extend)g(to)g(w)o(eigh) o(ts)g Ft(g)j Fv(of)d(b)q(ounded)h(v)m(ariation.)-108 2449 y(Generalising)d(from)f(in)o(terv)m(al)h(to)g(trees)g(could)g(ha)o (v)o(e)g(applications)g(to)h(the)f(study)g(of)h(t)o(w)o(o-dimensional)e (dynam-)-108 2509 y(ics,)20 b(as)h(w)o(e)f(no)o(w)g(explain.)32 b(In)20 b([9)o(],)h(Cvitano)o(vi)o(\023)-23 b(c,)19 b(Gunaratne,)j(and) e(Pro)q(caccia)g(in)o(tro)q(duced)g(the)g(concept)g(of)-108 2569 y Fq(pruning)e(fr)n(onts)e Fv(and)g(conjectured)g(that,)g (suitably)f(pruning)h(Smale's)e(horsesho)q(e)k(map)d(yields)f(to)j(a)f (complete)-108 2630 y(top)q(ological)h(description)f(of)h(the)f (dynamics)f(of)h(the)g(H)o(\023)-23 b(enon)16 b(family)e(|)j(this)f (they)g(called)f(the)h Fq(Pruning)i(F)l(r)n(ont)-108 2690 y(Conje)n(ctur)n(e)p Fv(.)j(De)14 b(Carv)m(alho)h([7])e(later)g (formalized)f(this,)i(giving)f(a)i(mathematical)10 b(de\014nition)k(of) g(pruning)g(fron)o(t)926 2819 y(2)p eop %%Page: 3 3 3 2 bop -108 286 a Fv(and)20 b(constructing)g(a)f(family)e Fr(P)23 b Fv(of)d(plane)f(homeomorphism)o(s)e(whic)o(h,)i (conjecturally)l(,)f(con)o(tains)h(the)g(H)o(\023)-23 b(enon)-108 347 y(family)13 b(up)h(to)h(semi-conjugacy)l(.)k(De)14 b(Carv)m(alho)i(also)f(sho)o(ws)g([8])f(that)g(the)h(family)d(P)i(is)g (in)g(1-1)i(corresp)q(ondence)-108 407 y(with)i(a)f(family)e(of)j(tree) e(and)i(dendrite)f(endomorphisms.)22 b(Being)16 b(able)h(to)h(compute)e (zeta)h(functions)g(for)h(tree)-108 467 y(endomorphisms)d(is)h(a)g (\014rst)h(step)f(to)o(w)o(ards)h(computing)e(the)h(zeta)g(functions)g (of)h(the)f(homeomorphisms)d(in)j(de)-108 527 y(Carv)m(alho's)22 b(family)d Fr(P)25 b Fv(and,)e(if)d(the)h(Pruning)h(F)l(ron)o(t)f (Conjecture)f(is)h(v)o(eri\014ed,)g(of)g(all)g(maps)f(in)h(the)g(H)o (\023)-23 b(enon)-108 587 y(family)l(.)-108 648 y(A)17 b(further)f(natural)h(step)g(will)f(consist)g(on)i(extending)e(our)h (results)f(to)h(in\014nite)f(trees,)g(whic)o(h)g(are)h(in)o(v)o(olv)o (ed)d(in)-108 708 y(the)i(de)g(Carv)m(alho's)i(family)c Fr(P)t Fv(.)-108 792 y(The)h(author)h(wishes)e(to)h(emphasize)e(that)i (once)f(the)g(address)i(is)e(de\014ned)g(one)h(pro)q(ceeds)g(v)o(ery)e (m)o(uc)o(h)g(along)i(the)-108 852 y(lines)f(of)g(and)h([2])f([5].)20 b(Man)o(y)14 b(thanks)g(to)h(Pierre)e(de)h(la)g(Harp)q(e,)h(Rob)o(yn)f (Curtis)g(and)h(Da)o(vid)f(Cimasoni)f(for)i(their)-108 912 y(remarks,)g(and)i(esp)q(ecially)e(to)h(Viviane)f(Baladi)h(for)h (her)f(help)f(and)i(patience)f(and)h(to)f(Andr)o(\023)-23 b(e)16 b(de)g(Carv)m(alho)i(for)-108 972 y(his)f(commen)n(ts.)-108 1199 y Fs(2)82 b(De\014nitions)25 b(and)i(statemen)n(t)g(of)g(results) -108 1323 y Ff(2.1)67 b(The)22 b(tree)g Fe(T)-108 1415 y Fv(Let)16 b Ft(T)21 b Fv(b)q(e)15 b(a)h(\014nite)e(tree,)g Ft(V)e Fv(\()p Ft(T)7 b Fv(\))14 b(its)h(set)g(of)g(v)o(ertices)e(and)j Ft(E)s Fv(\()p Ft(T)7 b Fv(\))14 b(its)h(set)g(of)h(edges.)21 b(Recalling)13 b(that)j(the)f(v)m(alence)-108 1475 y(of)i Ft(a)c Fr(2)h Ft(V)e Fv(\()p Ft(T)7 b Fv(\))15 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y Ft(f)-79 2272 y Fm(\016)p Fo(n)-21 2290 y Fv(is)k(single-v)m(alued)g(for)h(eac)o(h)e Ft(n)p Fv(\).)22 b(W)l(e)15 b(assume)h(for)g(the)g(momen)o(t)e(that)i(for)h (eac)o(h)f Ft(m)d Fr(\025)h Fv(1)i(the)g(set)g(Fix)p Ft(f)1884 2272 y Fm(\016)p Fo(m)1951 2290 y Fv(is)-108 2350 y(\014nite.)21 b(This)16 b(assumption)g(will)g(b)q(e)g(remo)o(v)o (ed)e(after)i(Lemma)e(3.2.)-108 2411 y(Let)e Ft(x)i Fr(2)e Fv(Fix)p Ft(f)163 2392 y Fm(\016)p Fo(m)213 2411 y Fv(,)g Ft(x)i Fr(2)g Ft(I)350 2418 y Fo(i)364 2411 y Fv(.)19 b(The)12 b(conditions)g(of)g(monotonicit)o(y)e(and)i(con)o(tin)o(uit)o (y)e(of)i(eac)o(h)f Ft(f)1560 2418 y Fo(i)1586 2411 y Fv(imply)e(that)k Ft(f)1844 2418 y Fo(i)1858 2411 y Fv(\()p Ft(I)1899 2418 y Fo(i)1912 2411 y Fv(\))h Fr(\032)-108 2471 y Ft(T)-79 2478 y Fo(i)-48 2471 y Fv(with)i Ft(T)92 2478 y Fo(i)122 2471 y Fv(homeomorphic)e(to)j(an)g(in)o(terv)m(al)e (and)i(con)o(taining)f Ft(a)1112 2478 y Fp(1)1132 2471 y Fv(.)22 b(\(This)16 b(follo)o(ws)g(from)g(the)g(imp)q(ossibilit)o(y)e (of)-108 2531 y(cases)20 b(suc)o(h)g(as)g(those)h(sho)o(wn)f(in)g (Figure)f(3\).)32 b(W)l(e)20 b(call)e Ft(T)994 2538 y Fo(i)1028 2531 y Fq(the)j(javelin)h(of)f Ft(f)1357 2538 y Fo(i)1371 2531 y Fv(.)32 b(Hence,)19 b Ft(f)1609 2513 y Fm(\016)p Fo(m)1680 2531 y Fq(ne)n(ar)h Ft(x)g Fv(can)g(b)q(e)-108 2591 y(regarded)d(as)g(a)g(map)e(of)h(the)g(in)o(terv)m(al)g(and)g(w)o (e)g(can)h(set)f(\(follo)o(wing)g([5]\))240 2694 y Ft(L)p Fv(\()p Ft(x;)8 b(f)371 2673 y Fm(\016)p Fo(m)422 2694 y Fv(\))14 b(=)g(0)j(if)e(the)h(graph)h(of)g Ft(f)899 2676 y Fm(\016)p Fo(m)967 2694 y Fv(do)q(esn't)f(cross)h(the)f (diagonal)h(at)g Ft(x)p Fv(.)926 2819 y(7)p eop %%Page: 8 8 8 7 bop -108 286 a Fv(If)16 b(the)g(graph)h(do)q(es)g(cross)g(the)f (diagonal)h(at)g Ft(x)p Fv(,)e(w)o(e)h(can)h(de\014ne)587 421 y Ft(L)p Fv(\()p Ft(x;)8 b(f)718 401 y Fm(\016)p Fo(m)769 421 y Fv(\))14 b(=)j(lim)854 451 y Fo(y)q Fm(!)p Fo(x)941 387 y Fv(sgn)q(\()p Ft(f)1060 369 y Fo(m)1093 387 y Fv(\()p Ft(y)r Fv(\))11 b Fr(\000)g Ft(y)r Fv(\))p 941 410 322 2 v 990 455 a(sgn)q(\()p Ft(x)g Fr(\000)f Ft(y)r Fv(\))1275 421 y Ft(;)-108 559 y Fv(where)16 b(sgn\()p Ft(x)p Fv(\))h(is)f(the)g(sign)h(of)f Ft(x)p Fv(.)21 b(W)l(e)16 b(also)h(set)414 714 y Ft(\027)s Fv(\()p Ft(x;)8 b(f)539 694 y Fm(\016)p Fo(m)590 714 y Fv(\))13 b(=)h Fr(\000)p Ft(L)p Fv(\()p Ft(x;)8 b(f)844 694 y Fm(\016)p Fo(m)895 714 y Fv(\))j Fr(\001)950 652 y Fo(m)p Fm(\000)p Fp(1)956 667 y Fl(Y)956 773 y Fo(k)q Fp(=0)1034 714 y Ft(\017)p Fv(\()p Ft(f)1102 694 y Fm(\016)p Fo(k)1141 714 y Ft(x)p Fv(\))22 b Fr(2)14 b(f)p Fv(0)p Ft(;)8 b Fv(1)p Ft(;)g Fr(\000)p Fv(1)p Fr(g)-108 869 y Fv(Let)15 b Ft(x)f Fr(2)g Fv(Fix)7 b(\()p Ft(f)194 851 y Fm(\016)p Fo(m)245 869 y Fv(\).)21 b(W)l(e)14 b(sa)o(y)g(that)h Ft(x)f Fv(is)h(attractiv)o(e)e(|)h(resp)q(ectiv)o(ely)e(repulsiv)o(e)h ({)i(if)f(there)f(exists)h(a)h(neigb)q(our-)-108 929 y(ho)q(o)q(d)j Ft(U)k Fv(of)17 b Ft(x)f Fv(suc)o(h)g(that)g Fr(8)p Ft(V)24 b Fr(\032)14 b Ft(U)5 b Fv(,)p Ft(x)13 b Fr(2)h Ft(V)28 b Fv(w)o(e)15 b(ha)o(v)o(e)h Ft(f)927 911 y Fm(\016)p Fo(m)978 929 y Fv(\()p Ft(V)11 b Fv(\))j Fr(\032)g Ft(V)27 b Fv(|)16 b(resp)q(ectiv)o(ely)e Ft(f)1537 911 y Fm(\016)p Fo(m)1588 929 y Fv(\()p Ft(V)e Fv(\))h Fr(\033)h Ft(V)28 b Fv(.)-108 1050 y Fu(Lemma)16 b(2.3:)-108 1110 y Fq(a\))k(If)g Ft(f)40 1092 y Fm(\016)p Fo(m)112 1110 y Fq(is)f(de)n(cr)n(e)n(asing)h(at)h Ft(x)d Fr(2)p Fq(Fix)p Ft(f)640 1092 y Fm(\016)p Fo(m)692 1110 y Fq(,)i(then)h(the)g (assumption)f(that)g Fv(#)p Fq(Fix)p Ft(f)1422 1092 y Fm(\016)p Fo(m)1492 1110 y Ft(<)f Fr(1)h Fq(for)f(al)r(l)j Ft(m)d Fq(implies)-108 1170 y(that)f Ft(x)f Fq(is)h(either)f(an)h(attr) n(acting)g(or)f(r)n(ep)n(el)r(ling)i(\014xe)n(d)f(p)n(oint)f(for)g Ft(f)1119 1152 y Fm(\016)p Fo(m)1170 1170 y Fq(.)-108 1230 y(b\))h Ft(\027)s Fv(\()p Ft(x;)8 b(f)77 1212 y Fm(\016)p Fo(m)128 1230 y Fv(\))14 b(=)g Fr(\000)p Fv(1)j Fr(\()-8 b(\))17 b Ft(f)431 1212 y Fm(\016)p Fo(m)500 1230 y Fq(is)g(incr)n(e)n(asing)h(and)g(attr)n(acting)g(at)f(the)h (\014xe)n(d)g(p)n(oint)g Ft(f)1502 1212 y Fm(\016)p Fo(m)1553 1230 y Fv(\()p Ft(x)p Fv(\))13 b(=)h Ft(x)p Fq(.)-108 1374 y Fv(The)j(pro)q(of)g(of)g(this)f(lemm)o(a)e(is)i(left)f(to)i(the) f(reader.)-108 1458 y(W)l(e)f(extend)g(no)o(w)g(the)g(set)g(Fix)p Ft(f)492 1440 y Fm(\016)p Fo(m)558 1458 y Fv(to)g(a)h(set)f(Fix)p Ft(?f)856 1440 y Fm(\016)p Fo(m)922 1458 y Fv(obtained)g(b)o(y)g (adding)g(some)f(sym)o(b)q(ols)g Ft(x)p Fr(\003)h Fv(where)g Ft(x)e Fr(2)h Ft(T)-108 1518 y Fv(and)j Fr(\003)f Fv(is)h(+,)e Fr(\000)p Fv(,)h(or)g(+)311 1525 y Fo(j)346 1518 y Fv(for)g(some)g Ft(j)j Fv(if)d Ft(x)d Fr(2)h Ft(V)d Fv(\()p Ft(T)c Fv(\))16 b(:)136 1661 y(Fix)10 b Ft(?)h(f)282 1640 y Fm(\016)p Fo(m)348 1661 y Fv(=)i(Fix)o Ft(f)499 1640 y Fm(\016)p Fo(m)559 1613 y Fl([)623 1590 y(\032)660 1661 y Ft(x)p Fr(\003)727 1588 y Fl(\014)727 1618 y(\014)727 1648 y(\014)727 1678 y(\014)778 1628 y Ft(f)807 1610 y Fm(\016)p Fo(m)858 1628 y Fv(\()p 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Fv(]])p Ft(:)-108 2549 y Fv(If)14 b Ft(f)-32 2531 y Fm(\016)p Fo(m)19 2549 y Fv(\()p Ft(a)64 2556 y Fo(i)78 2549 y Fr(\007)117 2556 y Fo(k)138 2549 y Fv(\))g Fr(6)p Fv(=)g Ft(a)249 2556 y Fo(i)276 2549 y Fv(for)h(all)e Ft(i)h Fv(and)h(all)e Ft(m)h Fr(\025)f Fv(1,)i(w)o(e)e(ma)o(y)g(replace)g(Fix)6 b Ft(?)h(f)1239 2531 y Fm(\016)p Fo(m)1304 2549 y Fv(b)o(y)14 b(Fix)o Ft(f)1470 2531 y Fm(\016)p Fo(m)1535 2549 y Fv(in)g(the)g(ab)q (o)o(v)o(e)g(form)o(ula.)-34 2609 y(Denote)j(b)o(y)f(P)o(er)25 b(the)17 b(set)g(of)g(p)q(erio)q(dic)g(orbits)h(of)f Ft(f)22 b Fv(whic)o(h)17 b(do)g(not)h(con)o(tain)f(an)o(y)g(of)g(the)g Ft(a)1678 2616 y Fo(i)1692 2609 y Fv(,)g(and)g(b)o(y)g(P)o(er)o Ft(?)-108 2669 y Fv(the)f(extended)g(set)g(of)g(p)q(erio)q(dic)g (orbits)h(formed)e(of)h(elemen)o(ts)e(of)i Fr([)1148 2676 y Fo(m)1182 2669 y Fv(Fix)o Ft(?f)1306 2651 y Fm(\016)p Fo(m)1358 2669 y Fv(.)926 2819 y(8)p eop %%Page: 9 9 9 8 bop -108 286 a Fv(If)15 b Ft(\015)i Fr(2)d Fv(P)o(er)p Ft(?)h Fv(is)g(of)h(p)q(erio)q(d)g Ft(p)p Fv(\()p Ft(\015)s Fv(\))g(\(i.e.,)d Ft(p)p Fv(\()p Ft(\015)s Fv(\))h(=)g(min)n Fr(f)p Ft(m)g Fr(\025)f Fv(1)23 b(:)e Ft(\015)c Fr(\032)d Fv(Fix)o Ft(?f)1283 268 y Fm(\016)p Fo(m)1335 286 y Fr(g)p Fv(\))h(and)h(con)o(tains)f(the)g(elemen)o(t)d Ft(\030)1961 293 y Fo(\015)-108 347 y Fv(w)o(e)k(let)662 447 y Ft(Z)t Fv(\()p Ft(\015)s Fv(\))e(=)831 381 y Fo(p)p Fp(\()p Fo(\015)r Fp(\))p Fm(\000)p Fp(1)854 399 y Fl(Y)854 506 y Fo(k)q Fp(=0)950 447 y Ft(g)r Fv(\()p Ft(f)1023 426 y Fm(\016)p Fo(k)1062 447 y Ft(\030)1083 454 y Fo(\015)1106 447 y Fv(\))22 b Fr(2)1189 437 y Fn(j)1179 447 y Fv(C)-108 571 y(With)16 b(this)h(notation,)f(it)g(is)g(not)h(di\016cult)e(to)h(v) o(erify)f(that)h(w)o(e)g(ha)o(v)o(e)719 673 y Ft(\020)740 680 y Fo(f)r(;g)789 673 y Fv(\()p Ft(t)p Fv(\))e(=)946 626 y Fl(Y)910 743 y Fo(\015)r Fm(2)p Fv(P)o(er)p Fo(?)1053 673 y Ft(F)7 b Fv(\()p Ft(\015)s Fv(\))-108 830 y(where)447 916 y Ft(F)g Fv(\()p Ft(\015)s Fv(\))13 b(=)h(exp)721 853 y Fm(1)702 868 y Fl(X)700 973 y Fo(m)p Fp(=1)790 882 y Ft(t)808 864 y Fo(m)p 790 904 51 2 v 794 950 a Ft(m)846 916 y(\027)s Fv(\()p Ft(\030)913 923 y Fo(\015)935 916 y Ft(;)8 b(f)986 895 y Fm(\016)p Fo(m)p Fm(\001)p Fo(p)p Fp(\()p Fo(\015)r Fp(\))1113 916 y Fv(\))p Ft(Z)t Fv(\()p Ft(\015)s Fv(\))1235 895 y Fo(m)1290 916 y Fr(2)1332 905 y Fn(j)1322 916 y Fv(C[[)p Ft(t)p Fv(]])-108 1041 y(Let)17 b Ft(\015)g Fr(2)d Fv(P)o(er)8 b(,)15 b(with)h Ft(p)p Fv(\()p Ft(\015)s Fv(\))f(=)e Ft(p)p Fv(,)k(and)g Ft(x)624 1048 y Fo(\015)659 1041 y Fr(2)d Ft(\015)s Fv(,)i(then)-108 1101 y Fu(\(a\))h Ft(F)7 b Fv(\()p Ft(\015)s Fv(\))13 b(=)g(1)k(if)f(the)g(graph)h(of)g Ft(f)542 1083 y Fm(\016)p Fo(p)596 1101 y Fv(do)q(es)g(not)f(cross)h(the)f(diagonal)h(at)g Ft(x)1279 1108 y Fo(\015)1301 1101 y Fv(;)-108 1161 y Fu(\(b\))f Ft(F)7 b Fv(\()p Ft(\015)s Fv(\))14 b(=)209 1141 y Fp(1)p 158 1150 119 2 v 158 1178 a(1)p Fm(\000)p Fo(Z)r Fp(\()p Fo(\015)r Fp(\))298 1161 y Fv(if)i Ft(f)372 1143 y Fm(\016)p Fo(p)426 1161 y Fv(is)g(increasing)g(at)h Ft(x)791 1168 y Fo(\015)829 1161 y Fv(and)g Ft(\015)i Fv(rep)q(elling;)-108 1228 y Fu(\(c\))d Ft(F)7 b Fv(\()p Ft(\015)s Fv(\))13 b(=)h(1)e Fr(\000)e Ft(Z)t Fv(\()p Ft(\015)s Fv(\))17 b(if)e Ft(f)425 1210 y Fm(\016)p Fo(p)479 1228 y Fv(is)h(increasing)g(at)h Ft(x)844 1235 y Fo(\015)882 1228 y Fv(and)g Ft(\015)i Fv(attracting;)-108 1288 y Fu(\(d\))d Ft(F)7 b Fv(\()p Ft(\015)s Fv(\))14 b(=)f(1)f(+)f Ft(Z)t Fv(\()p Ft(\015)s Fv(\))16 b(if)g Ft(f)431 1270 y Fm(\016)p Fo(p)485 1288 y Fv(is)g(decreasing)g(at)g Ft(x)857 1295 y Fo(\015)896 1288 y Fv(and)g Ft(\015)k Fv(attracting;)-108 1348 y Fu(\(e\))c Ft(F)7 b Fv(\()p Ft(\015)s Fv(\))13 b(=)203 1329 y Fp(1)p 153 1337 V 153 1366 a(1)p Fm(\000)p Fo(Z)r Fp(\()p Fo(\015)r Fp(\))293 1348 y Fv(if)j Ft(f)367 1330 y Fm(\016)p Fo(p)421 1348 y Fv(is)g(decreasing)g(at)g Ft(x)793 1355 y Fo(\015)832 1348 y Fv(and)g Ft(\015)k Fv(rep)q(elling.)-108 1427 y(Let)d Ft(\015)g Fr(2)d Fv(P)o(er)o Ft(?)e Fr(n)f Fv(P)o(er)c(,)16 b(then)-108 1487 y Fu(\(f)5 b(\))17 b Ft(F)7 b Fv(\()p Ft(\015)s Fv(\))13 b(=)g(1)k(if)f Ft(\015)j Fv(is)d(one-sided)g(rep)q (elling;)-108 1548 y Fu(\(g\))g Ft(F)7 b Fv(\()p Ft(\015)s Fv(\))13 b(=)h(1)e Fr(\000)f Ft(Z)t Fv(\()p Ft(\015)s Fv(\))16 b(if)f Ft(\015)20 b Fv(is)c(one-sided)g(attracting.)-34 1668 y(Notice)e(that)i(the)g(cases)g Fu(\(a\))f Fv(and)i Fu(\(d\))e Fv(where)h Ft(\015)i Fv(is)e(rep)q(elling)e(giv)o(e)h(the)h (same)e(expression)i(for)g Ft(F)7 b Fv(\()p Ft(\015)s Fv(\).)20 b(If)15 b(all)-108 1728 y(p)q(erio)q(dic)h(p)q(oin)o(ts)g (are)g(rep)q(elling,)e(and)i(if)f Ft(f)685 1710 y Fm(\016)p Fo(m)737 1728 y Ft(a)763 1735 y Fo(i)776 1728 y Fr(\003)f(6)p Fv(=)g Ft(a)893 1735 y Fo(i)907 1728 y Fv(,)h(for)h(all)f Ft(m)f Fr(\025)f Fv(1)j(and)h(0)d Fr(\024)f Ft(i)h Fr(\024)g Ft(N)5 b Fv(,)15 b(then)h(w)o(e)f(reco)o(v)o(er)f(the)-108 1788 y(usual)j(\(w)o(eigh)o(ted\))e(dynamical)g(zeta)h(function)g(of)g (the)g(tree)g(map)f Ft(f)5 b Fv(.)-108 1931 y Ff(2.6)67 b(Milnor-Th)n(urston)25 b(relation)f(for)e(tree)g(maps)-108 2023 y Fv(W)l(e)d(refer)e(to)i(Lemma)d(3.2)j(b)q(elo)o(w)f(for)h(a)g (de\014nition)f(of)h Ft(\020)970 2030 y Fo(f)r(;g)1018 2023 y Fv(\()p Ft(t)p Fv(\))f(where)h Ft(f)1265 2005 y Fm(\016)p Fo(m)1334 2023 y Fv(can)g(ha)o(v)o(e)f(in\014nitely)e(man)o (y)h(\014xed)-108 2083 y(p)q(oin)o(ts.)-108 2204 y Fu(Theorem)g(1:)k Fq(With)c(the)h(ab)n(ove)g(notations)h(and)e(assumptions)h(we)g(have)g (:)600 2301 y(a\))125 b Ft(\020)791 2308 y Fo(f)r(;g)839 2301 y Fv(\()p Ft(t)p Fv(\)\001)936 2275 y Fp(\(1\))936 2314 y Fo(f)r(;g)984 2301 y Fv(\()p Ft(t)p Fv(\))14 b(=)f(1)p Ft(;)542 b Fv(\(2\))602 2388 y Fq(b\))42 b Ft(\020)707 2395 y Fo(f)r(;g)756 2388 y Fv(\()p Ft(t)p Fv(\)\001)853 2362 y Fp(\(2\))853 2402 y Fo(f)r(;g)901 2388 y Fv(\()p Ft(t)p Fv(\))13 b(=)h(1)d(+)g Ft(r)1128 2395 y Fo(g)1148 2388 y Fv(\()p Ft(T)c Fv(\))p Ft(;)449 b Fv(\(3\))-108 2485 y Fq(wher)n(e)505 2555 y Ft(r)527 2562 y Fo(g)547 2555 y Fv(\()p Ft(T)7 b Fv(\))14 b(=)691 2521 y(1)p 691 2543 25 2 v 691 2589 a(2)782 2507 y Fl(X)729 2615 y Fo(a)748 2619 y Fk(n)769 2615 y Fm(2)p Fo(V)814 2621 y Fn(\025)p Fg(2)854 2615 y Fp(\()p Fo(T)5 b Fp(\))907 2555 y Fv(\()p Fq(val)q Fv(\()p Ft(a)1031 2562 y Fo(n)1054 2555 y Fv(\))11 b Fr(\000)g Fv(2\))p Ft(\017)1197 2562 y Fo(n)p Fm(\000)p Fp(1)1265 2555 y Ft(g)1288 2562 y Fo(n)p Fm(\000)p Fp(1)1357 2555 y Ft(;)-108 2694 y Fq(with)18 b Ft(V)26 2701 y Fm(\025)p Fp(2)74 2694 y Fv(\()p Ft(T)7 b Fv(\))13 b(=)h Fr(f)p Ft(a)264 2701 y Fo(n)287 2694 y Fr(j)p Ft(v)r(al)q Fv(\()p Ft(a)414 2701 y Fo(n)435 2694 y Fv(\))g Fr(\025)g Fv(2)p Fr(g)p Fq(.)926 2819 y Fv(9)p eop %%Page: 10 10 10 9 bop -108 286 a Fs(3)82 b(Pro)r(ofs)-108 410 y Ff(3.1)67 b(Chec)n(king)23 b(the)f(Milnor-Th)n(ur)q(ston)j(form)n(ula)e(on)f(a)g (particular)j(case)-108 503 y Fv(W)l(e)c(use)g(the)g(homotop)o(y)f (argumen)o(t)g(from)g([11])h(mo)q(di\014ed)f(in)h([5].)35 b(First,)21 b(let)f(us)h(c)o(hec)o(k)f(the)g(form)o(ula)g(on)h(a)-108 563 y(`simple')14 b(example)g(:)-108 623 y Fu(Lemma)i(3.1:)k Fq(L)n(et)d Ft(f)312 605 y Fp(0)346 623 y Fv(=)d(\()p Ft(f)446 605 y Fp(0)441 636 y(1)466 623 y Ft(;)8 b(:)g(:)g(:)f(;)h(f) 604 605 y Fp(0)599 636 y Fo(N)t Fm(\000)p Fp(1)678 623 y Fv(\))17 b Fq(b)n(e)g(a)g(pie)n(c)n(ewise)h(c)n(ontinuous)g(and)f (pie)n(c)n(ewise)h(monotone)g(multival-)-108 683 y(ue)n(d)c(tr)n(e)n(e) f(map)g(as)g(b)n(efor)n(e.)21 b(Supp)n(ose)13 b(further)g(that)h Ft(f)852 665 y Fp(0)847 696 y Fo(i)872 683 y Fv(\()p Ft(I)913 690 y Fo(i)926 683 y Fv(\))p Fq(is)f(c)n(ontaine)n(d)h(within) g(interior)f(of)h Ft(I)1602 690 y Fp(1)1621 683 y Fq(,)g(for)f Ft(i)g Fv(=)h(1)p Ft(;)8 b(:)g(:)g(:)g(;)g(N)-108 744 y Fq(and)18 b(that)g Ft(f)116 726 y Fp(0)111 756 y(1)150 744 y Fv(:)13 b Ft(I)199 751 y Fp(1)232 744 y Fr(!)h Ft(T)24 b Fq(is)17 b(di\013er)n(entiable)i(with)e Fr(j)p Fv(\()p Ft(f)859 726 y Fp(0)854 756 y(1)879 744 y Fv(\))898 726 y Fm(0)910 744 y Fr(j)d Ft(<)f Fv(1)p Fq(.)23 b(Then)611 860 y(a\))129 b Ft(\020)806 869 y Fo(f)827 859 y Fg(0)844 869 y Fo(;g)874 860 y Fv(\()p Ft(t)p Fv(\)\001)971 835 y Fp(\(1\))971 874 y Fo(f)r(;g)1019 860 y Fv(\()p Ft(t)p Fv(\))13 b(=)h(1)p Ft(;)612 931 y Fq(b\))43 b Ft(\020)718 939 y Fo(f)739 930 y Fg(0)756 939 y Fo(;g)786 931 y Fv(\()p Ft(t)p Fv(\)\001)883 906 y Fp(\(2\))883 945 y Fo(f)r(;g)931 931 y Fv(\()p Ft(t)p Fv(\))13 b(=)h(1)e(+)f Ft(r)1159 938 y Fo(g)1179 931 y Fv(\()p Ft(T)c Fv(\))p Ft(;)-108 1043 y Fq(wher)n(e)18 b Ft(r)52 1050 y Fo(g)72 1043 y Fv(\()p Ft(T)7 b Fv(\))17 b Fq(is)g(de\014ne)n(d)i(in)e(The)n(or)n(em)g (1.)-108 1115 y Fu(Pro)r(of:)k Fv(Our)16 b(assumptions)f(imply)e(that)j (the)f(set)h(of)g(p)q(erio)q(dic)f(p)q(oin)o(ts)h(of)f Ft(f)1311 1097 y Fp(0)1347 1115 y Fv(consists)h(of)g(a)g(single)f (\014xed)g(p)q(oin)o(t)-108 1175 y Ft(x)j Fv(whic)o(h)f(is)g(attractiv) o(e)g(\(the)g(orbit)h(of)g(eac)o(h)f(p)q(oin)o(t)h(in)f Ft(I)955 1182 y Fp(1)992 1175 y Fv(con)o(v)o(erges)g(to)h Ft(x)p Fv(\).)25 b(The)18 b(\014xed)f(p)q(oin)o(t)h Ft(x)f Fv(is)h(either)f(of)-108 1235 y(t)o(yp)q(e)f Fu(\(b\))g Fv(\(if)g Ft(\017)177 1242 y Fp(1)210 1235 y Fv(=)e(+1)i(\))h(or)f(of)h (t)o(yp)q(e)f Fu(\(c\))f Fv(\(if)h Ft(\017)769 1242 y Fp(1)802 1235 y Fv(=)e Fr(\000)p Fv(1)i(\).)22 b(Therefore,)15 b(in)h(either)f(case)h(w)o(e)g(ha)o(v)o(e)734 1345 y Ft(\020)755 1353 y Fo(f)776 1344 y Fg(0)793 1353 y Fo(;g)823 1345 y Fv(\()p Ft(t)p Fv(\))d(=)h(1)d Fr(\000)g Ft(\017)1049 1352 y Fp(1)1068 1345 y Ft(g)1091 1352 y Fp(1)1112 1345 y Ft(t:)-108 1455 y Fv(Let)17 b(us)f(c)o(hec)o(k)f(a\))h(\014rst,)h (i.e.)j(the)c(form)o(ula)e(for)j(the)f(\014rst)g(address.)-108 1515 y(With)g(the)g(\014rst)h(address)g(and)g(the)f(second)g(notation)h (of)g(Section)f(2.2,)g(w)o(e)g(ha)o(v)o(e)f(:)129 1661 y Ft(\022)q Fv(\()p Ft(a)198 1668 y Fo(k)219 1661 y Fv(+)257 1668 y Fo(l)270 1661 y Fv(\))e(=)h(\()p Ft(:)8 b(:)g(:)g(;)g Fv(+1)p Ft(;)g(:)g(:)g(:)f(;)h Fv(+1)p Ft(;)g Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)p Fv(\)+)p Ft(\017)936 1668 y Fp(\()p Fo(k)q(;l)p Fp(\))1005 1661 y Ft(g)1028 1668 y Fp(\()p Fo(k)q(;l)p Fp(\))1098 1661 y Ft(t)p Fr(\001)1147 1598 y Fm(1)1130 1613 y Fl(X)1133 1718 y Fo(n)p Fp(=1)1201 1661 y Fv(\()p Ft(\017)1240 1668 y Fp(1)1259 1661 y Ft(g)1282 1668 y Fp(1)1302 1661 y Fv(\))1321 1640 y Fo(n)p Fm(\000)p Fp(1)1390 1661 y Ft(t)1408 1640 y Fo(n)p Fm(\000)p Fp(1)1476 1661 y Fr(\001)p Fv(\(1)p Ft(;)g Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)g(;) g Fr(\000)p Fv(1\))p Ft(;)-108 1814 y Fv(and)17 b(\(with)f(the)g (\014rst)h(notation\))129 1962 y Ft(\022)q Fv(\()p Ft(a)198 1969 y Fo(i)211 1962 y Fr(\000)p Fv(\))d(=)g(\()p Ft(:)8 b(:)g(:)f(;)h Fv(+1)p Ft(;)g Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)g(;)g Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)o Fv(\)+)p Ft(\017)917 1969 y Fo(i)p Fm(\000)p Fp(1)976 1962 y Ft(g)999 1969 y Fo(i)p Fm(\000)p Fp(1)1058 1962 y Ft(t)p Fr(\001)1108 1900 y Fm(1)1090 1915 y Fl(X)1093 2019 y Fo(n)p Fp(=1)1162 1962 y Fv(\()p Ft(\017)1201 1969 y Fp(1)1220 1962 y Ft(g)1243 1969 y Fp(1)1263 1962 y Fv(\))1282 1942 y Fo(n)p Fm(\000)p Fp(1)1351 1962 y Ft(t)1369 1942 y Fo(n)p Fm(\000)p Fp(1)1437 1962 y Fr(\001)p Fv(\(1)p Ft(;)g Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)f(;) h Fr(\000)p Fv(1\))p Ft(;)-108 2116 y Fv(whic)o(h)22 b(is)f(our)h(\014rst)g(collapse)g(of)g(notation.)38 b(Note)22 b(that)g(w)o(e)f(do)i(not)f(ha)o(v)o(e)f(\()p Fr(\000)p Fv(1)p Ft(;)8 b Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)f(;)h Fr(\000)p Fv(1\))22 b(in)f(the)h(sum)-108 2176 y(b)q(ecause)17 b(of)f(the)g(ghost)i(edge)e(that)h(w)o(e)e(added)i(to)g(the)f(left)f (of)i Ft(a)1086 2183 y Fp(1)1105 2176 y Fv(.)k(Hence,)15 b(if)g Ft(i)f Fv(=)g Ft(k)k Fv(w)o(e)e(ha)o(v)o(e)f(:)129 2305 y Ft(K)170 2312 y Fo(k)q(;l)212 2305 y Fv(\()p Ft(t)p Fv(\))f(=)f(\(0)p Ft(;)8 b(:)g(:)g(:)g(;)g Fv(0)p Ft(;)g Fv(1)p Ft(;)g(:)g(:)g(:)g(;)g Fv(1)p Ft(;)g Fv(0)p Ft(;)g(:)g(:)g(:)g (;)g Fv(0\)+)932 2272 y(1)p 932 2294 25 2 v 932 2339 a(2)961 2305 y(\()p Ft(\017)1000 2313 y Fp(\()p Fo(k)q(;l)p Fp(\))1070 2305 y Ft(g)1093 2313 y Fp(\()p Fo(k)q(;l)p Fp(\))1162 2305 y Fr(\000)p Ft(\017)1221 2312 y Fo(i)1235 2305 y Ft(g)1258 2312 y Fo(i)1272 2305 y Fv(\))p Fr(\001)1394 2272 y Ft(t)p 1310 2294 186 2 v 1310 2339 a Fv(1)k Fr(\000)e Ft(\017)1415 2346 y Fp(1)1435 2339 y Ft(g)1458 2346 y Fp(1)1478 2339 y Ft(t)1501 2305 y Fr(\001)p Fv(\(1)p Ft(;)e Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)f(;)h Fr(\000)p Fv(1\))p Ft(:)-108 2438 y Fv(The)16 b(n)o(um)o(b)q(er)f(of)h(1's)g(in)g (the)g(constan)o(t)g(term)e(of)j Ft(K)848 2445 y Fo(k)q(;l)906 2438 y Fv(is)f(exactly)f(the)h(n)o(um)o(b)q(er)e(of)i(v)o(ertices)e(b)q (et)o(w)o(een)i Ft(a)1830 2445 y Fo(k)1850 2438 y Fr(\000)g Fv(and)-108 2498 y Ft(a)-82 2505 y Fo(k)-61 2498 y Fv(+)-23 2505 y Fo(l)9 2498 y Fv(according)k(to)f(the)g(order)g(\\)p Fr(\036)p Fv(".)31 b(\(See)18 b(Figures)h(1)h(and)f(2\).)31 b(In)18 b(other)i(w)o(ords,)g(if)e Ft(a)1580 2505 y Fo(k)1601 2498 y Fv(+)1639 2505 y Fo(l)1671 2498 y Fv(en)o(ters)h(edge)g Ft(I)1953 2505 y Fo(j)1970 2498 y Fv(,)-108 2558 y(then)e(this)f(n)o (um)o(b)q(er)e(is)i(just)h(\()p Ft(j)d Fr(\000)d Ft(k)i Fv(+)e(1\).)-108 2618 y(Our)17 b(metho)q(d)e(will)g(b)q(e)i(as)g(follo) o(ws)f(:)-108 2678 y(Since)k(w)o(e)g(are)g(w)o(orking)g(with)g(the)g (determinan)o(t)f(of)h(the)g(matrix)f Ft(K)1207 2686 y Fp(\()p Fo(k)q(;l)p Fp(\))p Fo(;m)1318 2678 y Fv(,)i(w)o(e)e(are)i(p) q(ermitted)d(to)j(p)q(erm)o(ute)914 2819 y(10)p eop %%Page: 11 11 11 10 bop -108 286 a Fv(lines)16 b(and)g(to)g(add)h(or)f(substract)h (lines)e(\(ro)o(ws\))h(to)h(other)f(lines)f(\(ro)o(ws\).)21 b(W)l(e)16 b(will)f(use)h(this)f(in)h(order)g(to)g(obtain)-108 347 y(a)h(new)f(matrix)f Ft(K)237 329 y Fm(0)249 347 y Fv(\()p Ft(t)p Fv(\))g(with)i Ft(K)477 329 y Fm(0)488 347 y Fv(\()p Ft(t)p Fv(\))d(=)f(1)f Fr(\000)f Fv(\011\()p Ft(t)p Fv(\),)k(where)h(\011\()p Ft(t)p Fv(\))g(will)f(b)q(e)h(of)h (rank)f(one.)22 b(Hence)285 503 y(\001)326 511 y Fo(f)347 502 y Fg(0)364 511 y Fo(;g)394 503 y Fv(\()p Ft(t)p Fv(\))13 b(=)h(det)o(\(1)e Fr(\000)f Fv(\011\()p Ft(t)p Fv(\)\))i(=)865 441 y Fo(N)t Fm(\000)p Fp(1)867 456 y Fl(X)870 560 y Fo(n)p Fp(=0)941 503 y Fv(\()p Fr(\000)p Fv(1\))1042 482 y Fo(n)1066 503 y Fv(T)l(r)19 b Fr(^)1168 482 y Fo(n)1203 503 y Fv(\011\()p Ft(t)p Fv(\))14 b(=)f(1)f Fr(\000)f Fv(T)l(r\011\()p Ft(t)p Fv(\))-34 654 y(W)l(e)17 b(will)f(call)h Fq(jump)g Fv(of)h Ft(a)441 661 y Fo(k)462 654 y Fv(+)500 661 y Fo(l)530 654 y Fv(the)f(n)o(um)o(b)q(er)f(of)i(1's)f(in)g(the)g (constan)o(t)h(term)e(of)i Ft(K)1482 661 y Fo(k)q(;l)1524 654 y Fv(.)25 b(Note)17 b(that)h(the)f(jump)-108 714 y(of)g Ft(a)-26 721 y Fo(k)-5 714 y Fv(+)33 721 y Fp(1)69 714 y Fv(is)f(alw)o(a)o(ys)g(1.)22 b(W)l(e)16 b(no)o(w)g(pro)q(ceed)g (inductiv)o(ely)e(to)j(construct)f(\011\()p Ft(t)p Fv(\).)-108 834 y(Let)h(us)h(call)e(`\014rst)g(branc)o(h')h(the)f(subtree)h(of)g Ft(T)23 b Fv(obtained)17 b(b)o(y)g(w)o(alking)f(from)g Ft(a)1379 841 y Fp(0)1415 834 y Fv(to)h Ft(a)1501 841 y Fo(s)1536 834 y Fv(passing)h(only)f(through)-108 895 y(the)e Ft(a)1 902 y Fo(k)22 895 y Fv(+)60 902 y Fp(1)80 895 y Fv(,)f(and)i(stopping)f(at)g(the)g(\014rst)g(v)o(ertex)e Ft(a)810 902 y Fo(s)843 895 y Fv(of)i(v)m(alence)f(one.)21 b(F)l(or)15 b(the)f(ro)o(ws)h(asso)q(ciated)h(to)f(the)g(v)o(ertices) -108 955 y(of)h(the)g(\014rst)g(branc)o(h,)f(except)f Ft(a)483 962 y Fo(s)501 955 y Fv(,)i(the)f(constan)o(t)h(term)e(is)i (exactly)e(what)i(w)o(e)f(require,)f(and)j(the)e(diagonal)i(term)-108 1015 y(of)i(\011)e(is)h Fr(\000)100 995 y Fp(1)p 100 1003 18 2 v 100 1032 a(2)122 1015 y Fv(\()p Ft(\017)161 1022 y Fo(i)175 1015 y Ft(g)198 1022 y Fo(i)225 1015 y Fr(\000)12 b Fv(\()p Fr(\006)p Ft(\017)354 1022 y Fo(i)p Fm(\000)p Fp(1)412 1015 y Ft(g)435 1022 y Fo(i)p Fm(\000)p Fp(1)495 1015 y Fv(\)\))g Fr(\001)g Ft(t)g Fr(\001)684 995 y Fp(1)p 631 1003 124 2 v 631 1032 a(1)p Fm(\000)p Fo(\017)690 1037 y Fg(1)708 1032 y Fo(g)725 1037 y Fg(1)742 1032 y Fo(t)760 1015 y Fv(,)18 b(for)g Ft(i)e Fv(=)g(1)p Ft(;)8 b(:)g(:)g(:)g(;)g(s)k Fr(\000)g Fv(1.)27 b(The)17 b(sign)h Fr(\006)g Fv(is)g(alw)o(a)o(ys)f(+1)i(except)d(in)-108 1075 y(the)j(\014rst)h(column,)e(i.e.)29 b(except)18 b(for)h Ft(a)623 1082 y Fp(1)642 1075 y Fv(.)30 b(If)19 b(w)o(e)g(sum)f(those)i(diagonal)g(terms,)e(they)g(will)g(all)h(v)m (anish,)h(except)-108 1135 y Fr(\000)p Ft(\017)-49 1142 y Fp(1)-29 1135 y Ft(g)-6 1142 y Fp(1)33 1135 y Fv(b)q(ecause)f(of)h (the)f(sign,)h(and)g Fr(\000)622 1116 y Fp(1)p 622 1124 18 2 v 622 1153 a(2)644 1135 y Ft(\017)664 1142 y Fo(s)p Fm(\000)p Fp(1)727 1135 y Ft(g)750 1142 y Fo(s)p Fm(\000)p Fp(1)833 1135 y Fv(b)q(ecause)g(it)e(app)q(ears)j(only)e(once)g(in)g (the)g(sum.)29 b(Hence,)18 b(the)-108 1196 y(`in)o(termediate)13 b(trace')j(of)g(\011)g(up)h(to)f Ft(K)605 1203 y Fo(s)p Fm(\000)p Fp(1)p Fo(;)p Fp(1)713 1196 y Fv(is)g(\()p Fr(\000)p Ft(\017)840 1203 y Fp(1)859 1196 y Ft(g)882 1203 y Fp(1)913 1196 y Fr(\000)968 1176 y Fp(1)p 968 1184 V 968 1213 a(2)991 1196 y Ft(\017)1011 1203 y Fo(s)p Fm(\000)p Fp(1)1074 1196 y Ft(g)1097 1203 y Fo(s)p Fm(\000)p Fp(1)1161 1196 y Fv(\))11 b Fr(\001)g Ft(t)g Fr(\001)1327 1176 y Fp(1)p 1274 1184 124 2 v 1274 1213 a(1)p Fm(\000)p Fo(\017)1333 1218 y Fg(1)1351 1213 y Fo(g)1368 1218 y Fg(1)1385 1213 y Fo(t)-108 1261 y Fv(Let)16 b(us)h(no)o(w)f(consider)f (the)h(next)f(edge)h Ft(I)j Fv(after)c Ft(a)817 1268 y Fo(s)851 1261 y Fv(\(according)h(to)g(our)h(order\).)k Ft(I)c Fv(=)d Ft(I)1513 1268 y Fo(r)o(;)p Fp(2)1573 1261 y Fv(for)i(some)e(1)h Ft(<)e(r)j(<)d(s)p Fv(.)-108 1321 y(The)24 b(jump)e(of)i Ft(a)222 1328 y Fo(r)241 1321 y Fv(+)279 1328 y Fp(2)322 1321 y Fv(is)f Ft(s)16 b Fr(\000)g Ft(r)q Fv(,)26 b(b)q(ecause)d(the)h(n)o(um)o(b)q(er)e(of)h(v)o(ertices) f(b)q(et)o(w)o(een)h Ft(a)1467 1328 y Fo(r)1485 1321 y Fr(\000)h Fv(and)g Ft(a)1676 1328 y Fo(r)1694 1321 y Fv(+)1732 1328 y Fp(2)1776 1321 y Fv(is)f(exactly)-108 1381 y Ft(s)14 b Fr(\000)g Ft(r)q Fv(,)22 b(so)f(there)f(are)g(exactly) f Ft(s)14 b Fr(\000)g Ft(r)22 b Fv(1's)e(in)h(the)f(constan)o(t)h(term) e(of)h(\011)1271 1388 y Fo(r)o(;)p Fp(2)1316 1381 y Fv(.)34 b(The)20 b(diagonal)i(term)c(of)j(\011)1886 1388 y Fo(r)o(;)p Fp(2)1951 1381 y Fv(is)-103 1422 y Fp(1)p -103 1430 18 2 v -103 1458 a(2)-80 1441 y Fv(\()p Ft(\017)-41 1448 y Fo(r)o(;)p Fp(2)3 1441 y Ft(g)26 1448 y Fo(r)o(;)p Fp(2)82 1441 y Fr(\000)11 b Ft(\017)152 1448 y Fo(r)q Fm(\000)p Fp(1)216 1441 y Ft(g)239 1448 y Fo(r)q Fm(\000)p Fp(1)303 1441 y Fv(\))g Fr(\001)g Ft(t)g Fr(\001)470 1422 y Fp(1)p 417 1430 124 2 v 417 1458 a(1)p Fm(\000)p Fo(\017)476 1463 y Fg(1)493 1458 y Fo(g)510 1463 y Fg(1)527 1458 y Fo(t)545 1441 y Fv(.)21 b(Notice)15 b(that)i Ft(\017)859 1448 y Fo(r)o(;)p Fp(2)903 1441 y Ft(g)926 1448 y Fo(r)o(;)p Fp(2)985 1441 y Fv(=)d Ft(\017)1057 1448 y Fo(s)1075 1441 y Ft(g)1098 1448 y Fo(s)1133 1441 y Fv(in)i(our)g(second)h (notation.)-108 1506 y(In)f(order)h(to)f(obtain)h(\(0)p Ft(;)8 b(:)g(:)g(:)g(;)g Fv(0)p Ft(;)g(:)g(:)g(:)g(;)g Fv(0)p Ft(;)g Fv(1)p Ft(;)g Fv(0)p Ft(;)g(:)g(:)g(:)o Fv(\))17 b(from)654 1671 y(\(0)p Ft(;)8 b(:)g(:)g(:)f(;)h Fv(0)p Ft(;)901 1598 y Fo(s)p Fm(\000)p Fo(r)852 1626 y Fl(z)p 874 1626 35 6 v 35 w(}|)p 953 1626 V 35 w({)852 1671 y Fv(1)p Ft(;)g(:)g(:)g(:)g(;)g Fv(1)p Ft(;)g Fv(0)p Ft(;)g(:)g(:)g(:)g(;)g Fv(0\))p Ft(;)-108 1781 y Fv(w)o(e)14 b(will)g(subtract)h(from)e Ft(K)398 1788 y Fo(r)o(;)p Fp(2)457 1781 y Fv(all)h(of)h(the)f Ft(K)700 1788 y Fo(v)735 1781 y Fv(for)h Ft(v)g Fv(=)f Ft(r)9 b Fv(+)e(1)p Ft(;)h(:)g(:)g(:)g(;) g(s)f Fr(\000)g Fv(1.)21 b(The)14 b(new)h(diagonal)g(term)d(of)j Ft(K)1847 1763 y Fm(0)1866 1781 y Fr(\000)7 b Fv(1)15 b(is)229 1891 y Fr(\000)273 1871 y Fp(1)p 273 1879 18 2 v 273 1908 a(2)295 1891 y Ft(t)c Fr(\001)g Fv(\(\()p Ft(\017)407 1898 y Fo(s)425 1891 y Ft(g)448 1898 y Fo(s)478 1891 y Fr(\000)f Ft(\017)547 1898 y Fo(r)q Fm(\000)p Fp(1)611 1891 y Ft(g)634 1898 y Fo(r)q Fm(\000)p Fp(1)699 1891 y Fv(\))h Fr(\000)g Ft(\017)799 1898 y Fo(s)p Fm(\000)p Fp(1)862 1891 y Ft(g)885 1898 y Fo(s)p Fm(\000)p Fp(1)960 1891 y Fv(+)g Ft(\017)1029 1898 y Fo(s)p Fm(\000)p Fp(2)1092 1891 y Ft(g)1115 1898 y Fo(s)p Fm(\000)p Fp(2)1190 1891 y Fr(\000)f Fv(+)e Ft(:)g(:)g(:)j Fr(\000)g Ft(\017)1424 1898 y Fo(r)1443 1891 y Ft(g)1466 1898 y Fo(r)1496 1891 y Fv(+)g Ft(\017)1565 1898 y Fo(r)q Fm(\000)p Fp(1)1629 1891 y Ft(g)1652 1898 y Fo(r)q Fm(\000)p Fp(1)1716 1891 y Fv(\))149 1951 y(=)250 1931 y Fp(1)p 250 1940 V 250 1968 a(2)273 1951 y Fv(\()p Ft(\017)312 1958 y Fo(s)p Fm(\000)p Fp(1)375 1951 y Ft(g)398 1958 y Fo(s)p Fm(\000)p Fp(1)473 1951 y Fr(\000)g Ft(\017)543 1958 y Fo(s)561 1951 y Ft(g)584 1958 y Fo(s)602 1951 y Fv(\))g Fr(\001)g Ft(t)p Fv(.)-108 2064 y(So,)17 b(the)f(`in)o(termedi)o(ate)d(trace')j (up)g(to)h Ft(K)661 2071 y Fo(r)o(;)p Fp(2)722 2064 y Fv(is)f(\()p Fr(\000)p Ft(\017)849 2071 y Fp(1)868 2064 y Ft(g)891 2071 y Fp(1)923 2064 y Fr(\000)977 2044 y Fp(1)p 977 2053 V 977 2081 a(2)1000 2064 y Ft(\017)1020 2071 y Fo(s)1038 2064 y Ft(g)1061 2071 y Fo(s)1080 2064 y Fv(\))11 b Fr(\001)g Ft(t)f Fr(\001)1246 2044 y Fp(1)p 1193 2053 124 2 v 1193 2081 a(1)p Fm(\000)p Fo(\017)1252 2086 y Fg(1)1270 2081 y Fo(g)1287 2086 y Fg(1)1304 2081 y Fo(t)1322 2064 y Fv(.)-108 2129 y(Using)17 b(the)f(same)f(tric)o(k)f (\(i.e.)20 b(subtracting)d(the)f Ft(K)835 2136 y Fo(u;v)903 2129 y Fv(from)f Ft(K)1059 2136 y Fo(k)q(;l)1117 2129 y Fv(for)i Ft(u;)8 b(v)17 b Fv(suc)o(h)f(that)h Ft(a)1525 2136 y Fo(u)1547 2129 y Fv(+)1585 2136 y Fo(v)1622 2129 y Fv(is)f(`b)q(et)o(w)o(een')e Ft(a)1912 2136 y Fo(k)1933 2129 y Fv(+)1971 2136 y Fo(l)-108 2189 y Fv(and)23 b Ft(a)19 2196 y Fo(k)40 2189 y Fr(\000)p Fv(\),)f(w)o(e)f(arriv)o(e)f (to)i(the)f(last)h(v)o(ertice)e Ft(a)795 2196 y Fo(N)828 2189 y Fv(,)i(and)g(the)g(in)o(termediate)c(trace)j(up)h(to)g Ft(a)1636 2196 y Fo(N)t Fm(\000)p Fp(1)1736 2189 y Fv(is)f(\()p Fr(\000)p Ft(\017)1868 2196 y Fp(1)1887 2189 y Ft(g)1910 2196 y Fp(1)1945 2189 y Fr(\000)-103 2230 y Fp(1)p -103 2238 18 2 v -103 2267 a(2)-80 2250 y Ft(\017)-60 2257 y Fo(N)t Fm(\000)p Fp(1)18 2250 y Ft(g)41 2257 y Fo(N)t Fm(\000)p Fp(1)121 2250 y Fv(\))11 b Fr(\001)f Ft(t)h Fr(\001)287 2230 y Fp(1)p 234 2238 124 2 v 234 2267 a(1)p Fm(\000)p Fo(\017)293 2272 y Fg(1)311 2267 y Fo(g)328 2272 y Fg(1)345 2267 y Fo(t)-108 2318 y Fv(Because)18 b(w)o(e)g(added)h(a)g(ghost)h(edge)e(to)h(the)g(righ)o(t)f(of)h Ft(a)943 2325 y Fo(N)976 2318 y Fv(,)g(the)f(term)f Fr(\000)1258 2299 y Fp(1)p 1258 2307 18 2 v 1258 2336 a(2)1280 2318 y Ft(\017)1300 2325 y Fo(N)t Fm(\000)p Fp(1)1379 2318 y Ft(g)1402 2325 y Fo(N)t Fm(\000)p Fp(1)1499 2318 y Fv(v)m(anishes)i(when)g(w)o(e)f(sum)-108 2379 y(up)f(to)g Ft(N)k Fv(and)c(\014nally)f(w)o(e)f(obtain)i(:)399 2508 y(\001)440 2516 y Fo(f)461 2507 y Fg(0)478 2516 y Fo(;g)507 2508 y Fv(\()p Ft(t)p Fv(\))d(=)g(1)d Fr(\000)g Fv(T)l(r\011\()p Ft(t)p Fv(\))i(=)h(1)d(+)1055 2474 y Ft(\017)1075 2481 y Fp(1)1095 2474 y Ft(g)1118 2481 y Fp(1)1138 2474 y Ft(t)p 1012 2497 186 2 v 1012 2542 a Fv(1)h Fr(\000)f Ft(\017)1118 2549 y Fp(1)1137 2542 y Ft(g)1160 2549 y Fp(1)1180 2542 y Ft(t)1217 2508 y Fv(=)1354 2474 y(1)p 1273 2497 V 1273 2542 a(1)h Fr(\000)f Ft(\017)1379 2549 y Fp(1)1398 2542 y Ft(g)1421 2549 y Fp(1)1441 2542 y Ft(t)1464 2508 y(:)-108 2678 y Fv(No)o(w,)16 b(let)g(us)g(consider)g (b\),)g(i.e.,)e(the)i(second)h(address.)914 2819 y(11)p eop %%Page: 12 12 12 11 bop -108 286 a Fv(W)l(e)17 b(use)h(exactly)e(the)h(same)f(metho)q (d)h(:)23 b(W)l(e)17 b(consider)g(the)g(trace)g(of)g(a)h(matrix)e (\011\()p Ft(t)p Fv(\))g(with)h Ft(K)t Fv(\()p Ft(t)p Fv(\))f(=)f(1)d Fr(\000)g Fv(\011\()p Ft(t)p Fv(\).)-108 347 y(With)k(the)g(second)h(address,)f(and)h Ft(k)f Fv(=)e Ft(i)i Fv(w)o(e)g(ha)o(v)o(e)137 476 y Ft(K)178 483 y Fo(k)q(;l)221 476 y Fv(\()p Ft(t)p Fv(\))d(=)h(\(0)p Ft(;)8 b(:)g(:)g(:)g(;)g Fv(0)p Ft(;)g Fv(1)p Ft(;)g Fv(0)p Ft(;)g(:)g(:)g(:)g(;)g Fv(0\))j(+)829 442 y(1)p 829 465 25 2 v 829 510 a(2)859 476 y(\()p Ft(\017)898 483 y Fo(k)q(;l)939 476 y Ft(g)962 483 y Fo(k)q(;l)1016 476 y Fr(\000)g Ft(\017)1086 483 y Fo(i)1099 476 y Ft(g)1122 483 y Fo(i)1137 476 y Fv(\))g Fr(\001)1269 442 y Fv(1)p 1197 465 168 2 v 1197 510 a(1)g Fr(\000)g Ft(\017)1302 517 y Fp(1)1322 510 y Ft(g)1345 517 y Fp(1)1381 476 y Fr(\001)g Fv(\(1)p Ft(;)d Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)f(;)h Fr(\000)p Fv(1\))p Ft(:)-108 611 y Fv(W)l(e)18 b(do)h(not)f(need)g(to)h (add)f(or)h(substract)f(lines)g(as)g(in)g(a\),)h(b)q(ecause)f(the)g (constan)o(t)g(term)f(is)h(already)g(what)g(w)o(e)-108 671 y(require.)i(Hence,)380 751 y(\011)418 758 y Fo(k)q(;l)460 751 y Fv(\()p Ft(t)p Fv(\))14 b(=)586 717 y(1)p 586 740 25 2 v 586 785 a(2)616 751 y(\()p Ft(\017)655 758 y Fo(k)q(;l)697 751 y Ft(g)720 758 y Fo(k)q(;l)773 751 y Fr(\000)d Ft(\017)843 758 y Fo(i)857 751 y Ft(g)880 758 y Fo(i)894 751 y Fv(\))g Fr(\001)1026 717 y Fv(1)p 954 740 168 2 v 954 785 a(1)g Fr(\000)g Ft(\017)1059 792 y Fp(1)1079 785 y Ft(g)1102 792 y Fp(1)1138 751 y Fr(\001)g Fv(\(1)p Ft(;)d Fr(\000)p Fv(1)p Ft(;)g(:)g(:)g(:)f(;)h Fr(\000)p Fv(1\))p Ft(:)-108 863 y Fv(The)16 b(correcting)e(factor)h(1)9 b(+)g Ft(r)458 870 y Fo(g)478 863 y Fv(\()p Ft(T)e Fv(\))14 b(app)q(ears)j(b)q(ecause) e Ft(\017)945 870 y Fo(i)959 863 y Ft(g)982 870 y Fo(i)1011 863 y Fv(will)f(b)q(e)i(coun)o(ted)e(negativ)o(ely)g(v)m(al\()p Ft(a)1686 870 y Fo(i)1699 863 y Fv(\))9 b Fr(\000)g Fv(2)15 b(times)e(in)-108 924 y(the)j(trace)g(of)h(\011\()p Ft(t)p Fv(\).)k(\(Lo)q(ok)c(at)g(Figures)f(1)h(and)f(2,)h(more)d(details)i (can)h(only)f(b)q(e)g(confusing\).)329 b Fc(2)-108 1068 y Fu(Note)20 b(:)k Fv(The)18 b(reader)f(has)i(probably)f(seen)f(that)h (in)f(the)h(pro)q(of)g(of)g(a\))g(w)o(e)g(repro)q(duce)f(exactly)g(the) g(w)o(alk)g(that)-108 1128 y(w)o(e)i(used)g(to)g(de\014ne)g(the)g (order)g(on)g Ft(T)7 b Fv(.)29 b(Hence,)18 b(the)g(argumen)o(t)g(w)o (ould)h(b)q(e)g(the)g(same)f(if)g(w)o(e)h(tak)o(e)f(an)o(y)h(other)-108 1188 y(order)f(compatible)d(with)i(the)g(orien)o(tation)g(and)h(whic)o (h)f(can)g(b)q(e)h(pro)q(duced)g(b)o(y)e(a)i(w)o(alk)f(of)h(the)f(same) f(t)o(yp)q(e,)g(to)-108 1248 y(de\014ne)h(the)f(\014rst)g(address.)23 b(Suc)o(h)16 b(orders)h(are)f(those)h(for)g(whic)o(h,)e(for)i Ft(a)1215 1255 y Fo(k)1252 1248 y Fv(a)g(v)o(ertex)e(of)i(v)m(alence)e (greater)i(than)g(1,)-108 1308 y(the)f(p)q(oin)o(ts)h(\\just)g(greater) f(than)h Ft(a)549 1315 y Fo(k)570 1308 y Fv(")g(m)o(ust)d(b)q(e)j(on)g (an)f(edge)g(inciden)o(t)f(to)i Ft(a)1316 1315 y Fo(k)1337 1308 y Fv(.)-108 1513 y Ff(3.2)67 b(T)-6 b(ec)n(hnical)24 b(lemmas)-108 1605 y Fv(The)c(metho)q(d)f(to)h(pro)o(v)o(e)e(the)i (theorem)e(is)h(v)o(ery)f(simple,)g(let)h(us)h(see)f(\014rst)h(the)f (simpler)e(case)j(where)f Ft(T)26 b Fv(is)19 b(an)-108 1666 y(in)o(terv)m(al.)k(Let)17 b Ft(f)23 b Fv(b)q(e)17 b(a)h(piecewise)d(monotone)i(transformation)g(of)h(this)f(in)o(terv)m (al.)22 b(It)17 b(is)g(p)q(ossible)g(to)h(c)o(ho)q(ose)f(a)-108 1726 y(function)f Ft(f)112 1708 y Fp(0)149 1726 y Fv(as)h(in)f(Lemma)e (3.1)i(suc)o(h)g(that)713 1836 y Ft(f)742 1815 y Fo(\025)779 1836 y Fv(:=)d(\(1)f Fr(\000)f Ft(\025)p Fv(\))p Ft(f)1025 1815 y Fp(0)1056 1836 y Fv(+)g Ft(\025f)-108 1946 y Fv(is)16 b(piecewise)f(strictly)g(monotone)g(for)i(all)f Ft(\025)e Fr(2)g Fv([0)p Ft(;)8 b Fv(1].)21 b(W)l(e)16 b(already)g(pro)o(v)o(ed)f (the)h(Theorem)f(1)h(for)h Ft(f)1757 1928 y Fp(0)1777 1946 y Fv(.)k(The)16 b(k)o(ey)-108 2006 y(of)j(the)f(theorem)f(is)h(to) g(sho)o(w)h(that)g(when)f(w)o(e)g(push)g Ft(f)921 1988 y Fp(0)960 2006 y Fv(to)g Ft(f)24 b Fv(through)19 b Ft(f)1284 1988 y Fo(\025)1307 2006 y Fv(,)f(\(2\))h(and)f(\(3\))h(of)g(Theorem)d (1)j(still)-108 2066 y(hold.)27 b(\(See)18 b([5])f(and)i([11]\).)26 b(No)o(w,)18 b(let)f(us)i(lo)q(ok)f(at)h(an)f(arbitrary)g(tree)g Ft(T)7 b Fv(.)26 b(Recalling)16 b(that)j(the)f(ja)o(v)o(elin)e Ft(T)1913 2073 y Fo(i)1945 2066 y Fv(of)-108 2126 y Ft(f)-84 2133 y Fo(i)-50 2126 y Fv(is)j(a)g(subtree)g(of)g Ft(T)26 b Fv(homeomorphic)17 b(to)i(the)g(in)o(terv)m(al)f(and)i(con)o(taining) f Ft(f)1353 2133 y Fo(i)1367 2126 y Fv(\()p Ft(I)1408 2133 y Fo(i)1421 2126 y Fv(\))13 b Fr([)h(f)p Ft(a)1551 2133 y Fp(1)1570 2126 y Fr(g)p Fv(,)19 b(w)o(e)g(shall)g(pro)q(ceed) -108 2187 y(exactly)14 b(as)h(in)g([5])f(but)h(considering)f(the)h Ft(f)687 2194 y Fo(i)701 2187 y Fv('s)f(separately)l(,)g(and)i (`pushing')e(eac)o(h)g Ft(f)1421 2194 y Fo(i)1450 2187 y Fv(through)i(its)e(ja)o(v)o(elin.)19 b(F)l(rom)-108 2247 y(no)o(w)e(on,)g(the)f(di\013erences)f(b)q(et)o(w)o(een)h(\014rst) g(and)h(second)g(address)g(will)e(not)i(b)q(e)f(of)h(an)o(y)f(imp)q (ortance,)f(so)i(w)o(e)f(will)-108 2307 y(simply)f(ignore)i(them)e(and) i(sp)q(eak)g(of)g Fq(the)g Fv(kneading)g(matrix,)d Fq(the)j Fv(kneading)g(determinan)o(t,)d(etc.)21 b(T)l(o)d(simplify)-108 2367 y(notation,)f(and)f(when)g(it)f(is)h(not)g(misleading,)e(w)o(e)h (will)g(write)g(\001)1088 2374 y Fo(f)1111 2367 y Ft(;)8 b(\020)1154 2374 y Fo(f)1192 2367 y Fv(rather)16 b(than)h(\001)1493 2374 y Fo(f)r(;g)1541 2367 y Ft(;)8 b(\020)1584 2374 y Fo(f)r(;g)1632 2367 y Fv(,)16 b(throughout)h(the)-108 2427 y(rest)g(of)f(this)g(section.)-108 2499 y(It)d(is)h(con)o(v)o (enien)o(t)d(to)j(use)f(the)g Ft(C)485 2481 y Fp(1)518 2499 y Fv(top)q(ology)h(\(for)g(the)f(de\014nition)g(of)g(di\013eren)o (tiabilit)o(y)e(giv)o(en)h(in)h(2.2\).)21 b(F)l(or)14 b Ft(r)h Fv(=)e(0)-108 2559 y(or)i(1,)g(let)e Ft(P)109 2541 y Fo(r)142 2559 y Fv(=)h Ft(P)232 2541 y Fo(r)251 2559 y Fv(\()p Ft(N)r(;)8 b Fv(\()p Ft(a)378 2566 y Fo(i)392 2559 y Fv(\))p Ft(;)g Fv(\()p Ft(\017)472 2566 y Fo(i)486 2559 y Fv(\)\))14 b(denote)g(the)g(set)g(of)g Ft(N)5 b Fv(-tuples)15 b Ft(f)k Fv(=)14 b(\()p Ft(f)1243 2566 y Fp(1)1262 2559 y Ft(;)8 b(:)g(:)g(:)g(;)g(f)1396 2566 y Fo(N)1429 2559 y Fv(\))14 b(where)g(eac)o(h)g Ft(f)1733 2566 y Fo(i)1761 2559 y Fv(:)f Ft(I)1810 2566 y Fo(i)1838 2559 y Fr(!)g Ft(T)21 b Fv(is)-108 2620 y Ft(C)-69 2602 y Fo(r)-34 2620 y Fv(and)c(strictly)d(monotone)i(increasing)g(or)g (monotone)g(decreasing)f(according)i(to)f(whether)g Ft(\017)1685 2627 y Fo(i)1714 2620 y Fv(equals)g(+1)g(or)-108 2680 y Fr(\000)p Fv(1.)21 b(If)14 b Ft(r)i Fv(=)d(1)i(w)o(e)g(further)f(imp) q(ose)g(that)h Ft(f)694 2662 y Fm(0)689 2692 y Fo(i)706 2680 y Fv(\()p Ft(a)751 2687 y Fo(k)772 2680 y Fv(\))e(=)h(0)h(for)g Ft(a)994 2687 y Fo(k)1030 2680 y Fv(the)f(endp)q(oin)o(ts)h(of)g Ft(f)1411 2687 y Fo(i)1440 2680 y Fv(\(v)m(anishing)g(of)g(the)f (lefthand)914 2819 y(12)p eop %%Page: 13 13 13 12 bop -108 286 a Fv(and)19 b(righ)o(thand)g(limit)c(of)k(the)f (deriv)m(ativ)o(e)e(at)j(in)o(terv)m(al)e(endp)q(oin)o(ts\).)27 b(The)19 b Ft(d)1336 293 y Fo(r)1373 286 y Fv(distance)f(b)q(et)o(w)o (een)f Ft(f)24 b Fv(and)19 b Ft(g)h Fv(in)-108 347 y Ft(P)-70 329 y Fo(r)-31 347 y Fv(is)e(giv)o(en)g(b)o(y)h(the)f(sum)g (of)i(the)e Ft(C)597 329 y Fo(r)635 347 y Fv(distances)g(b)q(et)o(w)o (een)g(the)h Ft(f)1148 354 y Fo(i)1162 347 y Fv('s)g(and)g Ft(g)1334 354 y Fo(i)1349 347 y Fv('s.)29 b(As)19 b(already)f(men)o (tioned,)f(w)o(e)-108 407 y(view)g Ft(f)22 b Fv(and)c(its)f(iterates)g Ft(f)421 389 y Fm(\016)p Fo(m)489 407 y Fv(as)h(m)o(ultiv)m(alued)c (maps)j Ft(T)22 b Fr(!)15 b Ft(T)7 b Fv(.)23 b(F)l(or)18 b(the)f(de\014nition)f(of)i Ft(\020)1603 414 y Fo(f)1626 407 y Fv(,)f(w)o(e)g(require)e(that)-108 467 y(the)g(sets)f(Fix)8 b Ft(f)177 449 y Fm(\016)p Fo(m)242 467 y Fv(b)q(e)15 b(\014nite,)f(but)g(this)g(condition)h(after)f(Lemma)e(3.2.)21 b(Let)15 b Ft(P)1343 449 y Fo(r)1336 480 y(M)1390 467 y Fv(consist)g(of)f(those)h Ft(f)k Fr(2)14 b Ft(P)1856 449 y Fo(r)1890 467 y Fv(suc)o(h)-108 527 y(that)j Ft(f)27 509 y Fm(\016)p Fo(m)78 527 y Fv(\()p Ft(a)123 534 y Fo(i)137 527 y Fr(\003)p Fv(\))d Fr(6)p Fv(=)g Ft(a)273 534 y Fo(i)303 527 y Fv(whenev)o(er)h(1)f Fr(\024)f Ft(m)h Fr(\024)f Ft(M)5 b Fv(,)17 b(0)d Fr(\024)f Ft(i)h Fr(\024)g Ft(N)21 b Fv(and)c Fr(\003)d Fv(=)f Fr(\006)j Fv(\(or)h(+)1390 534 y Fo(j)1408 527 y Fv(\).)-108 648 y Fu(Lemma)k(3.2:)30 b Ft(P)252 629 y Fo(r)245 660 y(M)306 648 y Fq(is)22 b(an)g(op)n(en)f(subset)i(of)e Ft(P)803 629 y Fo(r)823 648 y Fq(.)34 b(If)22 b Ft(J)955 655 y Fo(M)t Fp(+1)1061 648 y Fq(is)f(the)h(ide)n(al)g(of)f(elements)j(of)d(or)n(der)g Fr(\025)g Ft(M)e Fv(+)14 b(1)-108 708 y Fq(in)-53 698 y Fn(j)-64 708 y Fv(C[[)p Ft(t)p Fv(]])p Fq(,)i(i.e.,)i(the)g(ide)n(al) f(gener)n(ate)n(d)h(by)g(p)n(olynomials)f(of)g(de)n(gr)n(e)n(e)h Ft(M)e Fv(+)11 b(1)p Fq(,)18 b(the)g(map)719 816 y Ft(f)i Fr(7!)13 b Ft(\020)847 823 y Fo(f)920 816 y Fv(\()p Fq(mo)n(d)o Ft(J)1053 823 y Fo(M)t Fp(+1)1138 816 y Fv(\))-108 923 y Fq(de\014ne)n(d)19 b(on)f(the)g(set)g Fr(f)p Ft(f)h Fr(2)14 b Ft(P)437 905 y Fp(0)430 936 y Fo(M)492 923 y Fv(:)21 b(#)p Fq(Fix)9 b Ft(f)675 905 y Fm(\016)p Fo(m)740 923 y Ft(<)k Fr(1)8 b Ft(;)17 b Fr(8)p Ft(m)12 b Fr(\025)h Fv(1)p Fr(g)18 b Fq(extends)h(to)f(a)f(lo)n(c)n(al)r(ly)h(c)n(onstant)h (map)647 1031 y Ft(P)685 1011 y Fp(0)678 1043 y Fo(M)732 1031 y Fr(!)790 1021 y Fn(j)780 1031 y Fv(C[[)p Ft(t)p Fv(]])p Ft(=)p Fv(\()p Ft(J)959 1038 y Fo(M)t Fp(+1)1053 1031 y Fr(\\)1092 1021 y Fn(j)1082 1031 y Fv(C[[)p Ft(t)p Fv(]]\))8 b Ft(:)-108 1139 y Fq(\()p Ft(J)-61 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/Times-BoldItalic ff 270.00 scf sf 7951 1351 m gs 1 -1 sc (f) col-1 sh gr /Times-BoldItalic ff 180.00 scf sf 8101 751 m gs 1 -1 sc (p) col-1 sh gr /Times-BoldItalic ff 180.00 scf sf 8101 1201 m gs 1 -1 sc (p) col-1 sh gr /Times-BoldItalic ff 180.00 scf sf 8026 1651 m gs 1 -1 sc (p) col-1 sh gr /Times-BoldItalic ff 270.00 scf sf 8176 901 m gs 1 -1 sc (\(x\)) col-1 sh gr /Times-BoldItalic ff 270.00 scf sf 8176 1351 m gs 1 -1 sc (\(x\)) col-1 sh gr /Times-BoldItalic ff 270.00 scf sf 8101 1876 m gs 1 -1 sc (\(x\)) col-1 sh gr /Times-BoldItalic ff 270.00 scf sf 2776 1276 m gs 1 -1 sc (f) col-1 sh gr /Times-BoldItalic ff 270.00 scf sf 2775 825 m gs 1 -1 sc (u) col-1 sh gr /Times-BoldItalic ff 270.00 scf sf 2700 1800 m gs 1 -1 sc (v) col-1 sh gr /Times-BoldItalic ff 270.00 scf sf 7950 900 m gs 1 -1 sc (u) col-1 sh gr /Times-BoldItalic ff 270.00 scf sf 7875 1875 m gs 1 -1 sc (v) col-1 sh gr $F2psEnd rs %%EndDocument endTexFig 321 1283 a Fv(Figure)15 b(7:)22 b(Di\013eren)o(t)15 b(situations)i(when)g Ft(f)k Fv(crosses)c(the)f(diagonal)-108 1480 y(with)d Ft(x)28 1487 y Fo(i)54 1480 y Fv(and)g Ft(x)173 1487 y Fo(i)p Fp(+1)244 1480 y Fv(t)o(w)o(o)f(consecutiv)o(e)f (endp)q(oin)o(ts)i(of)g(the)f(partition)g Ft(\037)p Fv(.)20 b(W)l(e)12 b(sa)o(y)g(that)h Ft(h)f Fv(is)h(of)f(b)q(ounded)i(v)m (ariation)-108 1540 y(if)i(v)m(ar\()p Ft(h)p Fv(\))h(is)f(\014nite.) -108 1600 y(So,)22 b(let)d Ft(g)76 1607 y Fo(i)111 1600 y Fv(:)i Ft(I)168 1607 y Fo(i)202 1600 y Fr(!)267 1590 y Fn(j)257 1600 y Fv(C)p Ft(;)f(i)h Fv(=)f(1)p Ft(;)8 b Fr(\001)g(\001)g(\001)h Ft(;)f(N)19 b Fr(\000)13 b Fv(1)21 b(b)q(e)f(functions)h(of)f(b)q(ounded)h(v)m(ariation.)34 b(Let)20 b Ft(g)j Fv(=)e(\()p Ft(g)1733 1607 y Fp(1)1753 1600 y Ft(;)8 b Fr(\001)g(\001)g(\001)g Ft(;)g(g)1886 1607 y Fo(N)t Fm(\000)p Fp(1)1965 1600 y Fv(\))-108 1660 y(b)q(e)21 b(the)g(m)o(ultiv)m(alued)d(map)i Ft(T)28 b Fr(!)558 1650 y Fn(j)548 1660 y Fv(C.)35 b(Let)21 b Ft(Z)k Fv(=)d Ft(Z)875 1667 y Fp(1)916 1660 y Fv(=)g Ft(E)s Fv(\()p Ft(T)7 b Fv(\).)34 b(By)20 b(abuse)h(of)g(notation,)i(w) o(e)d(de\014ne)g(v)m(ar\()p Ft(g)r Fv(\))-108 1720 y(as)-45 1683 y Fl(P)8 1720 y Fv(v)m(ar\()p Ft(g)116 1727 y Fo(i)131 1720 y Fv(\).)30 b(Let)20 b Ft(Z)j Fv(=)c Ft(Z)431 1727 y Fp(1)470 1720 y Fv(=)g Ft(E)s Fv(\()p Ft(T)7 b Fv(\).)30 b(F)l(or)20 b Ft(n)f Fr(\025)g Fv(2)h(let)e Ft(Z)1031 1727 y Fo(n)1074 1720 y Fv(b)q(e)i(the)f Ft(n)1260 1702 y Fo(th)1315 1720 y Fv(re\014nemen)o(t)e(of)i Ft(Z)24 b Fv(under)19 b Ft(f)5 b Fv(,)20 b(i.e.)29 b(:)-108 1781 y Ft(I)-86 1788 y Fp(0)-55 1781 y Fr(\\)11 b Ft(f)18 1763 y Fm(\000)p Fp(1)66 1781 y Ft(I)88 1788 y Fp(1)118 1781 y Fr(\\)g Ft(:)d(:)g(:)i Fr(\\)i Ft(f)304 1763 y Fm(\000)p Fo(n)p Fp(+1)400 1781 y Ft(I)422 1788 y Fo(n)p Fm(\000)p Fp(1)506 1781 y Fv(with)17 b Ft(I)640 1788 y Fo(j)671 1781 y Fr(2)d Ft(Z)t Fv(.)-108 1841 y(W)l(e)k(can)g(then)g (de\014ne)f Ft(T)360 1823 y Fp(\()p Fo(n)p Fp(\))429 1841 y Fv(b)o(y)g(setting)h(:)24 b Ft(E)s Fv(\()p Ft(T)792 1823 y Fp(\()p Fo(n)p Fp(\))842 1841 y Fv(\))17 b(=)f Ft(Z)965 1848 y Fo(n)1007 1841 y Fv(and)j Ft(V)11 b Fv(\()p Ft(T)1198 1823 y Fp(\()p Fo(n)p Fp(\))1248 1841 y Fv(\))18 b(to)g(b)q(e)g(the)g(set)g(of)g(endp)q(oin)o(ts)g(of)g(the)-108 1901 y(partition)f Ft(Z)129 1908 y Fo(n)152 1901 y Fv(.)22 b(It)15 b(is)h(imp)q(ortan)o(t)f(to)i(note)f(that)h(eac)o(h)e Ft(a)f Fr(2)g Ft(V)d Fv(\()p Ft(T)1083 1883 y Fp(\()p Fo(n)p Fp(\))1133 1901 y Fv(\))p Fr(n)p Ft(V)g Fv(\()p Ft(T)c Fv(\))16 b(is)g(of)g(v)m(alence)f(2)i(and)f(therefore,)f(all) -108 1961 y Ft(T)-72 1943 y Fp(\()p Fo(n)p Fp(\))-3 1961 y Fv(are)j(homeomorphic)e(to)i Ft(T)7 b Fv(.)26 b(Moreo)o(v)o(er,)17 b Ft(T)25 b Fv(and)18 b Ft(T)963 1943 y Fp(\()p Fo(n)p Fp(\))1014 1961 y Fv(,)g(as)h(sets,)f(are)g(the)g(same.)25 b(Hence,)17 b Ft(f)24 b Fv(can)18 b(also)h(b)q(e)-108 2021 y(considered)d(as)h(a)g(map)e Ft(T)376 2003 y Fp(\()p Fo(n)p Fp(\))440 2021 y Fr(!)f Ft(T)540 2003 y Fp(\()p Fo(n)p Fp(\))590 2021 y Fv(.)-108 2082 y(W)l(e)22 b(sa)o(y)g(that)g Ft(J)27 b Fv(is)22 b(a)g Ft(Z)t Fq(-homterval)h Fv(if)e Ft(J)27 b Fv(is)22 b(a)g(maximal)d(non)o(trivial)i(subtree)g(of)h Ft(T)29 b Fv(whic)o(h)21 b(is)h(closed)f(and)-108 2142 y(has)f(the)f(prop)q(ert)o(y)g(that)h Fr(8)p Ft(n)e Fr(\025)g Fv(1)i(there)e(is)h Ft(I)k Fr(2)c Ft(Z)866 2149 y Fo(n)909 2142 y Fv(with)g Ft(J)k Fr(\032)c Ft(I)t Fv(.)29 b(Notice)18 b(that)i(a)f Ft(Z)t Fv(-hom)o(terv)m(al)f(is)h(alw)o(a)o(ys)-108 2202 y(homeomorphic)14 b(to)j(an)g(in)o(terv)m(al)e(\(b)q(ecause)h(it)g (is)g(a)h(closed)f(connex)g(subset)g(of)h(an)f(edge\).)-108 2262 y(W)l(e)g(mak)o(e)f(the)h(additional)g(assumption)g(that)h Ft(g)h Fv(is)e(constan)o(t)h(on)g(eac)o(h)e Ft(Z)t Fv(-hom)o(terv)m(al) g(of)i Ft(f)5 b Fv(.)-108 2322 y(F)l(or)18 b Ft(n)d Fr(\025)f Fv(1)k(and)f Ft(I)i Fr(2)c Ft(Z)337 2329 y Fo(n)361 2322 y Fv(,)h(let)h Fr(f)p Ft(x)516 2329 y Fo(I)s(;m)576 2322 y Ft(;)8 b(m)15 b Fr(\025)f Fv(1)p Fr(g)i(\032)e Ft(I)21 b Fv(b)q(e)c(suc)o(h)g(that)g(lim)1222 2329 y Fo(m)p Fm(!1)1325 2322 y Fr(j)p Ft(g)r Fv(\()p Ft(x)1411 2329 y Fo(I)s(;m)1472 2322 y Fv(\))p Fr(j)e Fv(=)g(ess)h(inf)1705 2329 y Fo(I)1725 2322 y Fr(j)p Ft(g)r Fr(j)h Fv(and)g(suc)o(h)-108 2383 y(that)i(lim)68 2390 y Fo(m)p Fm(!1)172 2383 y Ft(g)r Fv(\()p Ft(x)244 2390 y Fo(I)s(;m)305 2383 y Fv(\))f(exists.)27 b(W)l(e)19 b(set)f Ft(g)692 2364 y Fp(\()p Fo(n)p Fp(\))761 2383 y Fv(to)h(b)q(e)g(the)f(m)o(ultiv)m(alued)e(function)i(whose)h (restriction)e(on)i(eac)o(h)-108 2443 y Ft(I)f Fr(2)c Ft(E)s Fv(\()p Ft(T)73 2425 y Fp(\()p Fo(n)p Fp(\))123 2443 y Fv(\))g(=)g Ft(Z)241 2450 y Fo(n)281 2443 y Fv(is)i(equal)g(to)g (the)g(constan)o(t)h(lim)867 2450 y Fo(m)p Fm(!1)970 2443 y Ft(g)r Fv(\()p Ft(x)1042 2450 y Fo(I)s(;m)1103 2443 y Fv(\).)-108 2503 y(Eac)o(h)c Ft(T)46 2485 y Fp(\()p Fo(n)p Fp(\))110 2503 y Fv(is)g(a)g(\014nite)f(tree,)h(and)h Ft(g)540 2485 y Fp(\()p Fo(n)p Fp(\))604 2503 y Fv(is)e(constan)o(t)i (on)f(the)g(edges)g(of)h Ft(T)1204 2485 y Fp(\()p Fo(n)p Fp(\))1254 2503 y Fv(.)20 b(Hence,)12 b(w)o(e)h(can)g(built)f(the)h (kneading)-108 2563 y(matrix)g(as)j(in)e(2.4)h(\(w)o(e)f(no)o(w)h (denote)g(it)f(b)o(y)g Ft(M)750 2538 y Fp(\()p Fo(r)q Fp(\))745 2579 y(\()p Fo(n)p Fp(\))797 2563 y Fv(\()p Ft(t)p Fv(\),)g(with)g Ft(r)h Fv(=)f(1)p Ft(;)8 b Fv(2)15 b(dep)q(ending)g(on)g(the)g(address)g(c)o(hosen\),)f(and)-108 2642 y(w)o(e)i(note)h(its)f(determinan)o(t)e(b)o(y)h(\001)524 2616 y Fp(\()p Fo(r)q Fp(\))524 2661 y Fo(f)r(;g)571 2651 y Fg(\()p Fk(n)p Fg(\))618 2642 y Fv(.)914 2819 y(18)p eop %%Page: 19 19 19 18 bop -108 286 a Fv(W)l(e)16 b(note)h(b)o(y)e Ft(\020)173 298 y Fo(f)r(;g)220 288 y Fg(\()p Fk(n)p Fg(\))267 286 y Fv(\()p Ft(t)p Fv(\))h(the)g(reduced)g(zeta)g(function)g(of)g Ft(f)22 b Fv(with)16 b(a)g(w)o(eigh)o(t)g Ft(g)1331 268 y Fp(\()p Fo(n)p Fp(\))1382 286 y Fv(,)g(and)g(b)o(y)g Ft(\020)1595 293 y Fo(f)r(;g)1644 286 y Fv(\()p Ft(t)p Fv(\))g(the)g(one)g(with)-108 347 y(w)o(eigh)o(t)g Ft(g)r Fv(.)21 b(Let)436 449 y Ft(\032)461 429 y Fp(\()p Fo(n)p Fp(\))526 449 y Fv(=)14 b Ft(\032)p Fv(\()p Ft(g)647 429 y Fp(\()p Fo(n)p Fp(\))698 449 y Fv(\))g(=)26 b(lim)783 479 y Fo(n)p Fm(!1)875 379 y Fl(\022)911 449 y Fv(sup)938 489 y Fo(x)999 377 y Fl(\014)999 407 y(\014)999 437 y(\014)999 467 y(\014)1029 387 y Fo(n)p Fm(\000)p Fp(1)1031 402 y Fl(Y)1030 508 y Fo(k)q Fp(=0)1104 449 y Ft(g)1129 429 y Fp(\()p Fo(n)p Fp(\))1180 449 y Fv(\()p Ft(f)1228 429 y Fm(\016)p Fo(k)1267 449 y Ft(x)p Fv(\))1328 377 y Fl(\014)1328 407 y(\014)1328 437 y(\014)1328 467 y(\014)1344 379 y(\023)1381 390 y Fp(1)p Fo(=n)-108 587 y Fv(and)c(\026)-29 b Ft(\032)14 b Fv(:=)g(lim)159 594 y Fo(n)p Fm(!1)262 587 y Ft(\032)287 569 y Fp(\()p Fo(n)p Fp(\))338 587 y Fv(.)21 b(Notice)15 b(that)i(for)f(eac)o(h)g Ft(n)p Fv(,)g(\(2\))h(and)f(\(3\))h(of)g (Theorem)d(1)j(hold.)-108 731 y Ff(4.2)67 b(Results)-108 824 y Fu(Theorem)17 b(2)i(:)46 b Fq(L)n(et)17 b Ft(f)23 b Fq(and)17 b Ft(g)j Fq(b)n(e)e(as)f(de\014ne)n(d)h(b)n(efor)n(e.)k (With)c(the)g(ab)n(ove)g(notation,)g(we)g(have)g(:)-34 944 y(\(1\))e(for)g(e)n(ach)h Ft(n)d Fr(\025)g Fv(1)p Fq(,)p Fv(\001)409 919 y Fp(\()p Fo(r)q Fp(\))409 963 y Fo(f)r(;g)456 954 y Fg(\()p Fk(n)p Fg(\))503 944 y Fv(\()p Ft(t)p Fv(\))i Fq(|)h(r)n(esp)n(e)n(ctively)g Ft(\020)917 955 y Fo(f)r(;g)964 946 y Fg(\()p Fk(n)p Fg(\))1011 944 y Fv(\()p Ft(t)p Fv(\))f Fq(|)h(is)g(analytic)g(|)g(r)n (esp)n(e)n(ctively)g(mer)n(omorphic)-108 1017 y(|)h(in)f(the)h(op)n(en) g(disk)g Ft(D)355 1024 y Fo(n)396 1017 y Fq(of)f(r)n(adius)g Fv(1)p Ft(=\032)673 999 y Fp(\()p Fo(n)p Fp(\))725 1017 y Fq(,)g(and)h(the)f(fol)r(lowing)j(r)n(elations)e(hold)f(:)561 1134 y(a\))129 b Ft(\020)756 1145 y Fo(f)r(;g)803 1135 y Fg(\()p Fk(n)p Fg(\))850 1134 y Fv(\()p Ft(t)p Fv(\)\001)947 1108 y Fp(\(1\))947 1153 y Fo(f)r(;g)994 1143 y Fg(\()p Fk(n)p Fg(\))1040 1134 y Fv(\()p Ft(t)p Fv(\))14 b(=)f(1)564 1227 y Fq(b\))42 b Ft(\020)669 1238 y Fo(f)r(;g)716 1228 y Fg(\()p Fk(n)p Fg(\))763 1227 y Fv(\()p Ft(t)p Fv(\)\001)860 1201 y Fp(\(2\))860 1246 y Fo(f)r(;g)907 1236 y Fg(\()p Fk(n)p Fg(\))953 1227 y Fv(\()p Ft(t)p Fv(\))13 b(=)h(1)d(+)g Ft(r)1180 1234 y Fo(g)1200 1227 y Fv(\()p Ft(T)c Fv(\))1685 1306 y(\(5\))-108 1416 y Fq(with)18 b Ft(r)20 1423 y Fo(g)40 1416 y Fv(\()p Ft(T)7 b Fv(\))17 b Fq(as)g(in)h(the)n(or)n(em)f (1.)-108 1476 y(Assume)22 b(additional)r(ly)f(that)g Ft(g)i Fq(is)e(c)n(ontinuous)g(at)g(e)n(ach)g(p)n(oint)g(of)f(Fix)9 b Fv(\()p Ft(f)1288 1458 y Fm(\016)p Fo(m)1339 1476 y Fv(\))21 b Fq(and)g(one-side)n(d)g(c)n(ontinuous)h(at)-108 1537 y(e)n(ach)c(p)n(oint)f(of)h(Fix)p Ft(?)p Fv(\()p Ft(f)324 1519 y Fm(\016)p Fo(m)375 1537 y Fv(\))p Fr(n)p Fq(Fix)9 b Fv(\()p Ft(f)545 1519 y Fm(\016)p Fo(m)596 1537 y Fv(\))17 b Fq(for)g(al)r(l)i Ft(m)14 b Fr(\025)f Fv(1)p Fq(.)23 b(Then)18 b(:)-34 1657 y(\(2\))e(The)g(se)n(quenc)n(e)i Fv(\001)383 1632 y Fp(\()p Fo(r)q Fp(\))383 1676 y Fo(f)r(;g)430 1666 y Fg(\()p Fk(n)p Fg(\))476 1657 y Fv(\()p Ft(t)p Fv(\))e Fq(of)g(analytic)h(functions)g(in)g Ft(D)1096 1664 y Fo(n)1136 1657 y Fq(c)n(onver)n(ges)g(to)f(a)g(function)i Fv(\001)1682 1632 y Fp(\()p Fo(r)q Fp(\))1682 1671 y Fo(f)r(;g)1730 1657 y Fv(\()p Ft(t)p Fv(\))p Fq(,)e(analytic)-108 1724 y(in)i(the)g(disk)g Ft(D)e Fr(\032)d Ft(D)281 1731 y Fo(n)323 1724 y Fq(of)k(r)n(adius)f Fv(1)p Ft(=)t Fv(\026)-28 b Ft(\032)q Fq(.)22 b(Betwe)n(en)e(analytic)e(functions)h(in)f Ft(D)h Fq(we)f(have)g(the)g(identities)h(:)606 1841 y(a\))130 b Ft(\020)802 1848 y Fo(f)r(;g)850 1841 y Fv(\()p Ft(t)p Fv(\)\001)947 1816 y Fp(\(1\))947 1855 y Fo(f)r(;g)995 1841 y Fv(\()p Ft(t)p Fv(\))13 b(=)h(1)609 1928 y Fq(b\))42 b Ft(\020)714 1935 y Fo(f)r(;g)763 1928 y Fv(\()p Ft(t)p Fv(\)\001)860 1903 y Fp(\(2\))860 1942 y Fo(f)r(;g)907 1928 y Fv(\()p Ft(t)p Fv(\))14 b(=)g(1)d(+)g Ft(r)1135 1935 y Fo(g)1155 1928 y Fv(\()p Ft(T)c Fv(\))1685 2001 y(\(6\))-108 2171 y(The)17 b(complete)d(statemen)o(t)g(of)j(the)f (theorem)e(in)i(the)g(case)h(of)f(an)h(in)o(terv)m(al)e(map)g Ft(f)22 b Fv(should)17 b(b)q(e)f(read)h(in)e([2])h(.)-108 2231 y(The)h(pro)q(of)g(consists)g(of)f(t)o(w)o(o)g(p)q(oin)o(ts)h(:) -108 2292 y(Firstly)c(w)o(e)h(ha)o(v)o(e)f(to)h(pro)o(v)o(e)f(\(5\).)21 b(This)13 b(has)i(b)q(een)f(already)f(done)i(in)e([5])g(for)h(in)o (terv)m(al)f(maps)g(and)i(Theorem)d(1)j(for)-108 2352 y(tree)h(maps.)22 b(F)l(or)17 b(the)g(details)f(ab)q(out)i(the)e (radius)h(of)g(con)o(v)o(ergence,)e(see)h([3])g(and)h([10)q(].)22 b(The)17 b(correcting)f(factor)-108 2412 y(app)q(ears)21 b(in)e(this)h(computation)f(\(as)h(in)f(Theorem)f(1\),)i(and)h(then)e (the)g(address)h(do)q(es)h(not)f(pla)o(y)f(an)o(y)g(further)-108 2479 y(role.)i(W)l(e)16 b(will)f(then)h(write)g Ft(M)465 2487 y Fp(\()p Fo(n)p Fp(\))516 2479 y Fv(\()p Ft(t)p Fv(\))g(b)q(esides)g Ft(M)807 2453 y Fp(\()p Fo(r)q Fp(\))802 2495 y(\()p Fo(n)p Fp(\))854 2479 y Fv(\()p Ft(t)p Fv(\).)-108 2544 y(Secondly)l(,)g(to)g(pro)o(v)o(e)g(\(6\),)g(w)o(e)g(m)o(ust)f (examinate)g(the)h(con)o(v)o(ergence.)k(F)l(or)c(this,)g(Baladi)g(uses) h(a)g(crucial)e(lemm)o(a)-108 2605 y(\(The)i(Key)e(lemm)o(a)f(in)i ([2]\))f(whic)o(h)g(giv)o(es)h(a)g(condition)g(on)h(a)f(sequence)f(of)h (matrices)f(of)h(increasing)g(size)f(under)-108 2665 y(whic)o(h)f(the)g(sequence)f(of)i(their)e(determinan)o(ts)g(is)h(con)o (v)o(ergen)o(t)e(in)i(a)h(suitable)f(disk.)20 b(The)14 b(condition)g(is)g(that)h(the)914 2819 y(19)p eop %%Page: 20 20 20 19 bop -108 287 a Fv(sum)16 b(in)g(eac)o(h)h(column)e(of)i Ft(M)438 295 y Fp(\()p Fo(n)p Fp(\))489 287 y Fv(\()p Ft(t)p Fv(\))f(is)h(uniformly)d(b)q(ounded.)24 b(Let)16 b(us)h(note)g(b)o(y)f Ft(m)1423 262 y Fp(\()p Fo(n)p Fp(\))1423 300 y Fo(i;j)1474 287 y Fv(\()p Ft(t)p Fv(\))g(the)h(co)q (e\016cien)o(t)e(in)h(the)-108 347 y Ft(i)p Fv(-th)h(ro)o(w)f(and)h (the)f Ft(j)s Fv(-th)h(column)d(of)j Ft(M)635 355 y Fp(\()p Fo(n)p Fp(\))686 347 y Fv(\()p Ft(t)p Fv(\).)k(W)l(e)16 b(w)o(ould)g(lik)o(e)e(to)j(ha)o(v)o(e)e(the)h(follo)o(wing)g (inequalit)o(y)f(:)129 450 y Ft(m)172 424 y Fp(\()p Fo(n)p Fp(\))172 462 y Fo(i;j)222 450 y Fv(\()p Ft(t)p 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Fv(,)g(226-244)i(\(1995\).)129 1983 y([3])24 b(V.)k(Baladi)h(and)i(G.)e(Keller,)i Fq(Zeta)f(functions)h(and)f(tr)n (ansfer)f(op)n(er)n(ators)f(for)205 2043 y(pie)n(c)n(ewise)20 b(monotone)g(tr)n(ansformations,)e Fv(Comm.)e(Math.)i(Ph)o(ys.)g Fu(127)p Fv(,)h(459-477)205 2103 y(\(1990\).)129 2200 y([4])24 b(V.)d(Baladi,)j(A.)e(Kitaev,)h(D.)f(Ruelle,)h(and)g(S.)g (Semme)o(s,)e Fq(Sharp)j(determinants)205 2260 y(and)15 b(kne)n(ading)i(op)n(er)n(ators)c(for)i(holomorphic)g(maps,)30 b Fv(Pro)q(c.)14 b(Steklo)o(v)f(Math.)g(Inst.)205 2320 y Fu(216)p Fv(,)i(186-228)k(\(1997\).)129 2417 y([5])24 b(V.)e(Baladi)h(and)h(D.)g(Ruelle,)f Fq(A)o(n)h(extension)i(of)e(the)h (the)n(or)n(em)e(of)h(Milnor)g(and)205 2477 y(Thurston)f(on)h(the)f (zeta)h(functions)h(of)e(interval)i(maps)p Fv(,)e(Ergo)q(dic)h(Theory)e (Dy-)205 2537 y(namical)14 b(Systems)h Fu(14)p Fv(,)h(621-632)35 b(\(1994\).)129 2634 y([6])24 b(V.)17 b(Baladi)g(and)i(D.)e(Ruelle,)g Fq(Sharp)h(determinants,)39 b Fv(In)o(v)o(en)o(t.)16 b(Math.)h Fu(123)p Fv(,)h(553-)205 2694 y(574)f(\(1996\).)914 2819 y(20)p eop %%Page: 21 21 21 20 bop 129 286 a Fv([7])24 b(A.)16 b(de)h(Carv)m(alho,)h Fq(Pruning)h(fr)n(onts)f(and)h(the)f(formation)g(of)h(horsesho)n(es.)d Fv(T)l(o)i(ap-)205 347 y(p)q(ear)e(in)g(Ergo)q(dic)h(Theory)f(and)h (Dynamical)e(Systems.)129 448 y([8])24 b(A.)e(de)h(Carv)m(alho,)j Fq(Pruning)f(fr)n(onts)e(and)h(Thurston)g(classi\014c)n(ation)h(of)f (surfac)n(e)205 509 y(home)n(omorphisms)p Fv(.)14 b(In)i(preparation.) 129 610 y([9])24 b(P)l(.)e(Cvitano)o(vi)o(\023)-23 b(c,)24 b(G.H.)e(Gunaratne,)k(I.)c(Pro)q(caccia,)50 b Fq(T)l(op)n(olo)n(gic)n (al)24 b(and)g(metric)205 670 y(pr)n(op)n(erties)17 b(of)i(H)o(\023)-24 b(enon-typ)n(e)20 b(str)n(ange)f(attr)n(actors.)e Fv(Ph)o(ys.)g(Rev.)g (A)g(\(3\))h Fu(38)p Fv(,)g(no.)g(3,)205 731 y(1503{1520)h(\(1988\).) 129 832 y([10])24 b(F.)e(Hofbauer)i(and)f(G.)g(Keller,)g Fq(Zeta-functions)j(and)d(tr)n(ansfer-op)n(er)n(ators)f(for)205 892 y(pie)n(c)n(ewise)i(line)n(ar)g(tr)n(ansformations)p Fv(,)g(J.)e(Reine)g(Angew.)h(Math)g Fu(352)p Fv(,)i(100-113)205 953 y(\(1984\).)129 1054 y([11])f(J.)18 b(Milnor)f(et)h(W.)f(Th)o (urston,)i Fq(Iter)n(ate)n(d)g(maps)f(of)h(the)h(interval,)g Fv(in)d(Dynamical)205 1115 y(Systems)k(\(Maryland)i(1986-87\))i(,)f (Lecture)f(Notes)g(in)f(Math.)h(V)l(ol.)f(1342)i(J.C.)205 1175 y(Alexander)15 b(Springer-V)l(erlag,)g(Berlin)g(Heidelb)q(erg)f (New)i(Y)l(ork)g(\(1988\).)129 1276 y([12])24 b(D.)15 b(Ruelle,)e Fq(Sharp)k(zeta)g(functions)g(for)f(smo)n(oth)g(interval)i (maps)h Fv(In)o(ternat.)14 b(con-)205 1337 y(fer.)21 b(on)h(Dynamical)e(Systems)h(\(Mon)o(tevideo)g(1995\),)j(188-206)h(,)d (Pitman)f(Res.)205 1397 y(Notes)16 b(Math)g(362,)h(Longman)g(Harlo)o (w,)e(1996.)129 1499 y([13])24 b(D.)11 b(Ruelle,)f Fq(Dynamic)n(al)i (zeta)i(functions)f(for)g(maps)f(of)g(the)i(interval,)f Fv(Bull.)c(Amer.)205 1559 y(Math.)16 b(So)q(c.)g(\(N.S.\))f Fu(30)p Fv(,)h(212-214)i(\(1994\).)129 1660 y([14])24 b(D.)13 b(Ruelle,)f Fq(Dynamic)n(al)j(zeta)g(functions)h(for)e(pie)n(c) n(ewise)h(monotone)h(maps)f(of)f(the)205 1721 y(interval,)19 b Fv(CRM)e(Monograph)i(Series,)e(V)l(ol.4,)g(Amer.)e(Math.)i(So)q(c.,)g (Pro)o(vidence,)205 1781 y(\(1994\).)914 2819 y(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF