3. A NEIGHBORHOOD OF THE UNIT CIRCLE

Größe: px

Ab Seite anzeigen:

Download "3. A NEIGHBORHOOD OF THE UNIT CIRCLE"

Magdalena Heintze
vor 6 Jahren
Abrufe

1 3. A NEIGHBORHOOD OF THE UNIT CIRCLE Objekttyp: Chapter Zeitschrift: L'Enseignement Mathématique Band (Jahr): 39 (1993) Heft 3-4: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 (2.18) To show that fm, n (z) has a zéro P near a = a m,«, let Then g (a) =0. Consider the circle z-a = (10/i)" g(z) > m/100, while (2.19) 1. On this circle so for m = o(n), by Rouché's theorem g(z) and fm,n(z) hâve the same number of zéros inside the circle, namely one. This proves the claim and answers the Conway-Parker question. D 3. A NEIGHBORHOOD OF THE UNIT CIRCLE In this section we prove that an open neighborhood of{z:\z\= 1, z & 1} is contained in W. Lemma 3.1. If BçC is compact, n 1, z\<l, and (3.1) then every élément of B is expressible in the form (3.2) In particular, if -1 eb, then ze W. Proof Given b m eb, inductively pick b m+leß and e m, e {o, l}, m >0, 1 < / < n such that Successive substitution yields

3 Since B is compact, z Mn b M ~^ OasM-^oo, so which is the désired form.? z - z2z Proposition 3.1. If zer, -l<z<-(p" 1, then ze W. Proof. Let B=[-1,- z]. Then, since -I<z - (f)" 1 implies 2-1, we hâve We now apply Lemma 3.1 with n=l, and conclude that zew.? Lemma 3.2. If BçC is compact, -leß, OUeC a/irf (3.3) where int 5 dénotes the interior of S, then there is a neighborhood N of x such that Proof. Condition (3.3) implies that (3.1) holds for zina neighborhood of x, so Lemma 3.2 follows from Lemma 3.1.? Lemma 3.3. // B= {z:\z\^r} for some R 1, n 1, x\=l and (3.4) then Proof. Since W contains the unit circle and is closed under z^> l/z, this follows trivially from Lemma 3.2.? Proposition 3.2. If \x\= 1, x± ±1, then xe int W.

4 l0)l Proof We claim that if R 2, then the condition (3.4) of Lemma 3.3 holds for n large enough. If x = exp(tua) and a is irrational, then by Kronecker's theorem {xx J :j 1} is dense on the unit circle, and then for every ô > 0, the disk of radius R ô is contained in the union on the right side of (3.4) for n large enough. If ais rational, then the {x :j J 1 } are the vertices of a regular k-gon, and k 3 since x ±I.ln that case the union on the right side of (3.4) contains a disk of radius r, where r, 1, and R are the sides of a triangle, and the angle between the sides of lengths r and 1 is n/k. Therefore, by the Law of Cosines, and so Since cos(n/k) cos(tt/3), we find that for R 2, since (R 2-3/4) 1/2 -Risan increasing function of R.? Proving - 1 e int W is trickier, because it will not do to take B as a dise of radius lif Im(z) is small compared to Re(z+ 1). We will instead take B as a parallelogram that becomes flatter and flatter as Imz -> 0. The following two lemmas will be used in verifying the condition of Lemma 3.1. LEMMA3.4. Let T=(~\ _J). Let uj = TJ( 0 ) =(- \)J( *). Then for n^\6 contains {(( a b ):a,b ez, a 1, b\ n- 16} Proof. coordinate a. Next e3e pick Given such ( ), first else pick l5 s2s 2 so that e^ + z2z 2 v 2 has first 3 =s4=oorlso that e^ + e2e 2 u 2 + 3^ ^4 has first coordinate a and second coordinate b' with b r b mod 2. Certainly b' <l+2+3+4= 10, so b-b' n-6.lf6>b\ then

5 If b < b\ then Lemma 3.5. Let T, Uj be as in Lemma 3.4. Let B be the square with vertices (± 1, ± 1). Then for n 35, Proof. \Tn is the parallelogram with vertices The cross-section of this with x-coordinate x 0 is the vertical interval [-nx - 1/2, - nx /2] for - 1/2 x0 1/2. Hence given (a, (3) eb pick ae{-1,0,1} such that - 1/2 a +-a 1/2 and then pick bez such that -n(a+a)+ 1/2 3 +b -n(a+a)+ 1/2. Since P <l and a+a 1/2, we see b\ \(n+l)+l n-16 if n 35. Then (a, (3) + (a, b) e\ TnßT that ey^y =- n B and by Lemma 3.4 we can pick 5i,...,8 n such (ût, Z?), so Lemma 3.5 follows.? Proposition e int W. Proof. Since Wis closed under z^> z and z 1/z, it suffices to show that for z < 1, Imz>o, and \z+ l sufficiently small, z is in PT. (Proposition 3.1 handles the case z e R.) Let ô = z + 1. Let 5 be the parallelogram with vertices ± 1 ± ô. We work in a nonstandard coordinate System for C, with basis vectors 1 and ô, so B is represented by the square with vertices (±l, ±1). We claim that multiplication by z is represented by the matrix T= (~\ _J) up to 0( 8 ). We hâve and so ô2ô 2 corresponds to ( ô 2, -2Reô)in our basis, and is 0( 8 )

6 From Lemma 3.5, it foliows then that so for sufficiently small ô, we may apply Lemma 3.1 to deduce zew.? We now combine ail the results of this section. Theorem 3.1. contained in W. There is an open neighborhood of {z: z =l,z^l} Proof. Apply Propositions 3.2 and 3.3.? Corollary 3.1. If ze(-1,-1+8) for sufficiently small ô then z is a multiple zéro of some 0, 1 power séries. Proof By Theorem 3. 1, if ôis small enough we can pick 0, 1 power séries /? and zéros z n of /? such that z n $R and zn ~~> Z as n-> 00. By taking a subsequence we may assume that the coefficient of z k in /? is eventually constant for large n, for each k. By a Rouché's Theorem argument, the pairs of zéros {z n,zn } of fn must converge to (at least) a double zéro at z of lima.? n?? oo 4. $\bar{w}$is CONNECTED Since W iscountable, we cannot hope to prove Wis connected. We prove instead that W is connected. First we need some topolbgical lemmas. Give {0, 1} the discrète topology and {o,l} the product topology, as usual. If v= (u 1, u 2,..., v n ) is a finite vector of o's and Ts, let S v be the set of séquences in {0, l} 03 which start with v. The following lemma is the key ingrédient in the connectivity proof. Lemma 4.1. Let Y be a topological space. Suppose f: {0, l} 03 -? Y is a continuous map such that (4.1) for ail ue{q,l} n, and ail n 0. (Hère vo dénotes the vector v with 0 appended, etc.) Then the image of f is path connected.

Ähnliche Dokumente

Kalkausscheidungen bei Flachdächern

Kalkausscheidungen bei Flachdächern Autor(en): Trüb, U. Objekttyp: Article Zeitschrift: Cementbulletin Band (Jahr): 34-35 (1966-1967) Heft 15 PDF erstellt am: 31.01.2017 Persistenter Link: http://doi.org/10.5169/seals-153465