Classification of topologically conjugate affine mappings
We consider affine mappings from $ℝ^n$ into $ℝ^n, n ≥ 1$. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from $ℝ$ into $ℝ$ and also...
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| Дата: | 2009 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3009 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509023830278144 |
|---|---|
| author | Budnyts'ka, T. V. Будницька, Т. В. |
| author_facet | Budnyts'ka, T. V. Будницька, Т. В. |
| author_sort | Budnyts'ka, T. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:07Z |
| description | We consider affine mappings from $ℝ^n$ into $ℝ^n, n ≥ 1$. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from $ℝ$ into $ℝ$ and also for affine mappings from $ℝ^n$ into $ℝ^n, n > 1$, having at least one fixed point and the nonperiodic linear part. |
| first_indexed | 2026-03-24T02:34:31Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 515.126, 517.91
T. V. Budnyc\ka (Ky]v. nac. un-t im. T. Íevçenka)
KLASYFIKACIQ TOPOLOHIÇNO SPRQÛENYX
AFINNYX VIDOBRAÛEN|
We investigate affine maps from Rn to Rn , n ≥ 1. We prove the theorem on topological conjugacy
of an affine map having at least one fixed point with corresponding linear map. We obtain the
classification up to topological conjugacy of affine maps from R to R and also of those affine maps
from Rn to Rn , n > 1, that have at least one fixed point and whose linear parts are not periodic.
Rassmatryvagtsq affynn¥e otobraΩenyq yz Rn v Rn , n ≥ 1. Dokazana teorema o topolohy-
çeskoj soprqΩennosty affynnoho otobraΩenyq, ymegweho xotq b¥ odnu nepodvyΩnug toçku,
s sootvetstvugwym lynejn¥m otobraΩenyem. Poluçena klassyfykacyq, s toçnost\g do topo-
lohyçeskoj soprqΩennosty, affynn¥x otobraΩenyj yz R v R, a takΩe tex affynn¥x otob-
raΩenyj yz Rn v Rn , n > 1, kotor¥e ymegt xotq b¥ odnu nepodvyΩnug toçku y ç\y lynej-
n¥e çasty ne qvlqgtsq peryodyçeskymy.
1. Vstup. U roboti rozhlqdagt\sq matryci nad polem dijsnyx çysel.
Podibni ( n × n ) -matryci A ta B budemo poznaçaty A l∼ B. Takym çynom,
dva linijnyx vidobraΩennq f, g : R
n → R
n nazyvagt\ linijno sprqΩenymy,
qkwo isnu[ bi[ktyvne linijne vidobraΩennq h : Rn → R
n take, wo g = h � f �
� h–
1 ( poznaçatymemo f l∼ g ).
Oznaçennq 1.1. VidobraΩennq f, g : Rn → R
n nazyvagt\ topolohiçno
sprqΩenymy ( i poznaçagt\ f t∼ g ), qkwo isnu[ homeomorfizm h : Rn → R
n
takyj, wo
g = h � f � h–
1.
Problema topolohiçno] klasyfikaci] linijnyx vidobraΩen\ z R
n v R
n davno
pryvertala uvahu matematykiv. Znaçnyj vklad u vyrißennq c\oho pytannq
zrobyly N. H. Kuiper ta J. W. Robbin [1, 2], qki klasyfikuvaly vsi neperiodyçni
linijni vidobraΩennq. Zadaçu topolohiçno] klasyfikaci] periodyçnyx vidobra-
Ωen\ çastkovo rozv�qzaly S. E. Cappell ta J. L. Shaneson [3 – 5] ta W. C. Hsiang ta
W. Pardon [6], I. Madsen ta M. Rothenberg [7], R. Schultz [8]. Ta, nezvaΩagçy na
ce, topolohiçna klasyfikaciq periodyçnyx linijnyx vidobraΩen\ we j s\ohodni
zalyßa[t\sq do kincq ne vyvçenog, a razom z cym zalyßa[t\sq ne rozv�qzanog
zadaça topolohiçno] klasyfikaci] vsix linijnyx vidobraΩen\.
Nexaj f : Rn → Rn , A — ( n × n ) -matrycq, b ∈ R
n � fiksovanyj vektor.
VidobraΩennq vyhlqdu f ( x ) = A x + b nazyvagt\ afinnym vidobraΩennqm, a
matrycg A � linijnog çastynog vidobraΩennq f.
Problema topolohiçno] klasyfikaci] afinnyx vidobraΩen\ zalyßa[t\sq vid-
krytog. DoslidΩenng c\oho pytannq j prysvqçeno danu stattg, v qkij otry-
mano klasyfikacig, z toçnistg do topolohiçno] sprqΩenosti, çastyny afinnyx
vidobraΩen\ z Rn v Rn , n > 1, a takoΩ vsix afinnyx vidobraΩen\ z R v R .
© T. V. BUDNYC|KA, 2009
134 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
KLASYFIKACIQ TOPOLOHIÇNO SPRQÛENYX AFINNYX VIDOBRAÛEN| 135
2. Topolohiçna klasyfikaciq linijnyx vidobraΩen\. Dlq klasyfikaci]
afinnyx vidobraΩen\, z toçnistg do topolohiçno] sprqΩenosti, vykorystovu[t\-
sq analohiçna klasyfikaciq linijnyx vidobraΩen\, tomu nahada[mo deqki vΩe vi-
domi rezul\taty.
Nexaj f : Rn → Rn � linijne vidobraΩennq. Todi R
n moΩna rozklasty v
prqmu sumu svo]x f-invariantnyx pidprostoriv:
R
n = W
+( f ) � W
–( f ) � W
∞( f ) � W
0( f ) .
Vyznaçymo vidobraΩennq fα = f W fα ( ) , α = +, –, ∞, 0, takym çynom:
VidobraΩennq Xarakterystyçni çysla λ
vidobraΩennq
f+ 0 < | λ | < 1
f– | λ | > 1
f∞ λ = 0
f0 | λ | = 1
Poznaçymo çerez dim ( fα ) rozmirnist\ W
α
( f ), a çerez or ( f ) znak vyznaçny-
ka matryci, wo vidpovida[ bi[ktyvnomu vidobraΩenng f ; qkwo or ( f ) = + 1, to f
zberiha[ ori[ntacig, qkwo or ( f ) = – 1, to f ori[ntacig ne zberiha[ [1, 2].
VidobraΩennq f : Rn → Rn nazyvagt\ periodyçnym, qkwo isnu[ k ∈ N take,
wo f k = id R n . Najmenße take çyslo k nazyva[t\sq periodom vidobraΩennq f.
Teorema 2.1 [1, 2] da[ topolohiçnu klasyfikacig neperiodyçnyx linijnyx vi-
dobraΩen\.
Teorema 2.1. Nexaj f, g : Rn → R
n � linijni vidobraΩennq, qki ne [ perio-
dyçnymy. VidobraΩennq f i g topolohiçno sprqΩeni todi i til\ky todi, koly
dim ( f+ ) = dim ( g+ ), or ( f+ ) = or ( g+ ), dim ( f– ) = dim ( g– ), or ( f– ) = or ( g– ), f∞ l∼ g∞,
f0 l∼ g0 .
Vidomo, wo qkwo f : Rn → Rn � linijne vidobraΩennq, to isnu[ bazys pros-
toru Rn , v qkomu matrycq vidobraΩennq f moΩe buty zvedena do dijsno] kano-
niçno] formy [9], a neobxidnog ta dostatn\og umovog podibnosti dvox matryc\ [
rivnist\ ]xnix dijsnyx kanoniçnyx form.
Tobto, qkwo f ( x ) = A x, to, zvivßy matrycg A do dijsno] kanoniçno] formy
A′, zhrupu[mo bloky ci[] matryci tak, wob v rezul\tati vona mala vyhlqd
A′ =
A
A
A
A
+
−
∞
0 0 0
0 0 0
0 0 0
0 0 0 0
,
de matryci Aα, α = +, –, ∞, 0, vyznaçeno takym çynom:
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
136 T. V. BUDNYC|KA
Matrycq
Xarakterystyçni çysla λ
matryci
A+ 0 < | λ | < 1
A– | λ | > 1
A∞ λ = 0
A0 | λ | = 1
Todi teoremu 2.1 moΩna sformulgvaty takym çynom.
Teorema 2.1′′′′. Nexaj f, g : Rn → Rn , de f ( x ) = A x, g ( x ) = C x � linijni vi-
dobraΩennq, qki ne [ periodyçnymy. VidobraΩennq f i g topolohiçno sprqΩeni
todi i til\ky todi, koly rank ( A+ ) = rank ( C+ ), sign ( det ( A+ ) ) = sign ( det ( C+ ) ),
rank ( A– ) = rank ( C– ), sign ( det ( A– ) ) = sign ( det ( C– ) ), A∞ = C∞ , A0 = C0 .
ZauvaΩennq 2.1. Varto naholosyty na tomu, wo, vykorystovugçy teore-
mu 2.1 (abo teoremu 2.1′ ), moΩna klasyfikuvaty lyße neperiodyçni vidobra-
Ωennq.
Takym çynom, topolohiçna klasyfikaciq vsix linijnyx vidobraΩen\ zvodyt\-
sq do klasyfikaci] periodyçnyx linijnyx vidobraΩen\. U robotax [1, 2] bulo
zrobleno prypuwennq, wo dlq klasu periodyçnyx linijnyx vidobraΩen\ topo-
lohiçna sprqΩenist\ oznaça[ linijnu sprqΩenist\, ta dovedeno joho dlq vypad-
kiv, koly period vidobraΩen\ s = 1, 2, 3, 4 abo 6 (tobto dlq matryc\, usi vlasni
çysla qkyx [ s-my korenqmy z 1). Pizniße bulo znajdeno j inßi periody ta
umovy, dlq qkyx ce prypuwennq bulo istynnym [3 – 8]. Ale S. E. Cappell ta
J. L. Shaneson [10], pobuduvavßy kontrpryklad, dovely xybnist\ takoho prypu-
wennq dlq dovil\noho periodu vidobraΩennq.
Topolohiçna klasyfikaciq periodyçnyx linijnyx vidobraΩen\ zalyßa[t\sq
ne zaverßenog, a otΩe, zalyßa[t\sq j ne rozv�qzanog zadaça topolohiçno] kla-
syfikaci] vsix linijnyx vidobraΩen\.
3. Topolohiçna klasyfikaciq afinnyx vidobraΩen\. Dlq krawoho rozu-
minnq suti zadaçi, wo vyvça[t\sq, rozhlqnemo spoçatku vypadok n = 1. Dlq li-
nijnyx vidobraΩen\ ma[ misce take tverdΩennq.
TverdΩennq 3.1 [1]. Nexaj f : R → R, f ( x ) = a x, de a ∈ R � linijne vido-
braΩennq.
Isnugt\ 7 klasiv topolohiçno sprqΩenyx linijnyx vidobraΩen\. Try klasy
vyznaçagt\sq çyslamy a = 0, 1, – 1, a inßi � vidkrytymy intervalamy miΩ
çyslamy 0, 1 ta – 1 na R.
Tobto qkwo f ( x ) = a x, g ( x ) = c x, de a, c ∈ R, to f t∼ g todi i til\ky todi,
koly a t a c abo odnoçasno naleΩat\ do odni[] z komponent R \ { 0, 1, – 1 },
abo odnoçasno dorivnggt\ 0, 1 abo – 1.
TverdΩennq 3.2 da[ neobxidni ta dostatni umovy topolohiçno] sprqΩenosti
dvox afinnyx vidobraΩen\ z R v R.
TverdΩennq 3.2. Nexaj f : R → R, f ( x ) = a x + b, de a, b ∈ R � afinne
vidobraΩennq.
Isnugt\ 8 klasiv topolohiçno sprqΩenyx afinnyx vidobraΩen\. Dva klasy
vyznaçagt\sq çyslamy a = 0, – 1, dva klasy � paramy çysel a = 1, b = 0 ta
a = 1, b ≠ 0, a inßi � vidkrytymy intervalamy miΩ çyslamy 0, 1 ta – 1
na R.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
KLASYFIKACIQ TOPOLOHIÇNO SPRQÛENYX AFINNYX VIDOBRAÛEN| 137
Tobto qkwo f ( x ) = a x + b, g ( x ) = c x + d, de a, b, c, d ∈ R, t o f t∼ g todi
i til\ky todi, koly a ta c abo odnoçasno naleΩat\ do odni[] z komponent
R \ { 0, 1, – 1 }, abo odnoçasno dorivnggt\ 0, 1 abo – 1. Qkwo a = c = 1, to b
ta d abo odnoçasno dorivnggt\ 0, abo odnoçasno vidminni vid 0.
Dovedennq. Topolohiçno sprqΩeni vidobraΩennq magt\ odnakovi topolo-
hiçni vlastyvosti. Takym çynom, navedemo ti topolohiçni vlastyvosti, qki roz-
dilqgt\ mnoΩynu vidobraΩen\ vyhlqdu f ( x ) = a x + b, de a, b ∈ R, na 8 klasiv
ta digt\ iz R v R :
1) f ( x ) = b ≡ const todi i til\ky todi, koly a = 0 ;
2) f � totoΩne vidobraΩennq todi i til\ky todi, koly a = 1 ta b = 0;
3) f ne ma[ neruxomyx toçok todi i til\ky todi, koly a = 1 ta b ≠ 0;
4) f 2 � totoΩne vidobraΩennq, a f ne [ totoΩnym vidobraΩennqm todi i
til\ky todi, koly a = – 1;
5) f zberiha[ ori[ntacig todi i til\ky todi, koly a > 0;
6) lim
n
nf x
b
a→∞
( ) =
−1
∈ R dlq vsix x ∈ R todi i til\ky todi, koly | a | < 1.
Vlastyvosti 5 ta 6 porodΩugt\ rozbyttq R na 4 intervaly, a vlastyvosti 1 �
4 � toçky 0, 1 ta – 1 miΩ nymy. Vraxovugçy te, wo çyslu a = 1 pry b = 0 ta
b ≠ 0 vidpovidagt\ dva riznyx klasy topolohiçno sprqΩenyx afinnyx vido-
braΩen\ (ce vyplyva[ z vlastyvostej 2 ta 3), v rezul\tati otrymu[mo 8 klasiv
topolohiçno sprqΩenyx afinnyx vidobraΩen\.
VidobraΩennq f ( x ) = a x + b, a ≠ 1, b ∈ R, ta g ( x ) = c x + d, c ≠ 1, d ∈ R, de
a ta c abo odnoçasno naleΩat\ do odni[] z komponent R \ { 0, 1, – 1 }, abo odno-
çasno dorivnggt\ 0 abo – 1, [ topolohiçno sprqΩenymy, bo isnu[ homeomor-
fizm h : R → R takyj, wo g = h � f � h–
1, de
h ( x ) =
d
c
x
b
a
x
b
a
x
b
a
d
c
a c x
b
a
l
l
1 1
1 1 1 1
1
−
=
−
−
−
−
−
+
−
= ≠
−
−
, ,
, , .
VidobraΩennq f ( x ) = x + b, b ≠ 0, ta g ( x ) = x + d , d ≠ 0, topolohiçno
sprqΩeni, oskil\ky isnu[ h ( x ) =
d
b
x m+ , m ∈ R, take, wo g = h � f � h–
1.
TverdΩennq 3.2 dovedeno.
ZauvaΩennq 3.1. Qkwo v tverdΩenni 3.2 rozhlqnuty vidobraΩennq f ( x ) =
= a x + b, de a ∈ R, b = 0, to otryma[mo tverdΩennq 3.1, wo [ cilkom pryrod-
nym, adΩe mnoΩyna vsix linijnyx vidobraΩen\ [ pidmnoΩynog vsix afinnyx vi-
dobraΩen\.
ZauvaΩymo, wo v monohrafi] [11] navedeno rozv�qzky funkcional\noho riv-
nqnnq h � f = g � h dlq ßyrokoho klasu vidobraΩen\ f, g : R → R, h � nevidome
vidobraΩennq z R v R.
U cij roboti rozpoçato topolohiçnu klasyfikacig afinnyx vidobraΩen\ z
R
n v Rn, n > 1, qka idejno [ skladnißog, niΩ u vypadku n = 1. Vona ©runtu[t\-
sq na isnuvanni neruxomyx toçok afinnoho vidobraΩennq. Dlq dovedennq osnov-
no] teoremy sformulg[mo deqki dopomiΩni rezul\taty.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
138 T. V. BUDNYC|KA
Toçku x ∈ R
n nazyvagt\ neruxomog toçkog vidobraΩennq f : Rn → R
n ,
qkwo f ( x ) = x.
Lema 3.1. Nexaj vidobraΩennq f, g : Rn → R
n taki, wo f ( x ) = A x + b,
g ( x ) = A x, F = { φ : Rn → Rn
| g = φ � f � φ–
1
}.
Qkwo f t∼ g, to isnu[ homeomorfizm h ∈ F takyj, wo h ( q ) = 0, de q �
neruxoma toçka vidobraΩennq f.
Dovedennq. VidobraΩennq f i g topolohiçno sprqΩeni, tomu isnu[ homeo-
morfizm ψ takyj, wo g = = ψ � f � ψ–
1 (tobto ψ–
1 � g = f � ψ–
1
).
Vyznaçymo h tak: y = h ( x ) = ψ ( x ) – ψ ( q ) (tomu h–
1
( y ) = ψ–
1
( y + ψ ( q ) ) ).
Rozhlqnemo h � f � h–
1
( y ) = h � f � ψ–
1
( y + ψ ( q ) ) = h � ψ–
1 � g ( y + ψ ( q ) ) =
= h ( ψ–
1 � g ( y + ψ ( q ) ) ) = ψ ( ψ–
1
[ g ( y + ψ ( q ) ) ] ) – ψ ( q ) = g ( y + ψ ( q ) ) – ψ ( q ) =
= g ( y ) + g � ψ ( q ) – ψ ( q ) = g ( y ) + ψ � f ( q ) – ψ ( q ) = g ( y ) + ψ ( q ) – ψ ( q ) = g ( y ).
Lemu 3.1 dovedeno.
Teorema 3.1 da[ neobxidni ta dostatni umovy dlq topolohiçno] sprqΩenosti
afinnoho vidobraΩennq ta vidpovidnoho linijnoho.
Teorema 3.1. Nexaj vidobraΩennq f, g : Rn → R
n taki, wo f ( x ) = A x + b,
g ( x ) = A x. VidobraΩennq f i g topolohiçno sprqΩeni todi i til\ky todi, koly
isnu[ q ∈ R
n take, wo f ( q ) = q.
Dovedennq. Neobxidnist\. VidobraΩennq f ( x ) = A x + b t∼ g ( x ) = A x, tomu
vidobraΩennq f ta g magt\ odnakovu kil\kist\ neruxomyx toçok.
Oskil\ky g ( 0 ) = 0, to g ( x ) = A x ma[ prynajmni odnu neruxomu toçku.
OtΩe, f ( x ) = A x + b teΩ ma[ prynajmni odnu neruxomu toçku, bo inakße f, g
ne topolohiçno sprqΩeni.
Dostatnist\. Za umovog isnu[ toçka q ∈ R
n taka, wo f ( q ) = q = A q + b.
Todi f t∼ g, bo isnu[ homeomorfizm h : Rn → Rn , h ( x ) = x + q, takyj, wo f = h �
� g � h–
1.
Teoremu 3.1 dovedeno.
Damo kryterij topolohiçno] sprqΩenosti afinnyx vidobraΩen\, wo magt\
xoça b po odnij neruxomij toçci ta linijni çastyny qkyx ne [ periodyçnymy.
Teorema 3.2. Nexaj f, g : Rn → Rn, de f ( x ) = A x + b, g ( x ) = C x + d � taki
vidobraΩennq, wo:
1) isnugt\ q, α ∈ R
n taki, wo f ( q ) = q ta g ( α ) = α;
2) ne isnu[ k, l ∈ N takyx, wo Ak = E, Cl = E.
VidobraΩennq f i g topolohiçno sprqΩeni todi i til\ky todi, koly rank ( A+ ) =
= rank ( C+ ), sign ( det ( A+ ) ) = sign ( det ( C+ ) ), rank ( A– ) = rank ( C– ), sign ( det ( A– ) ) =
= sign ( det ( C– ) ), A∞ = C∞ , A0 = C0 .
Dovedennq. Za umovog teoremy vidobraΩennq f ta g magt\ neruxomi toç-
ky q ta α vidpovidno. Tomu, vykorystovugçy teoremu 3.1, ma[mo
f ( x ) = A x + b t∼ r ( x ) = A x,
g ( x ) = C x + d t∼ s ( x ) = C x.
Zastosovugçy dlq vidobraΩen\ r ( x ) = A x ta s ( x ) = C x teoremu 2.1′, otry-
mu[mo neobxidnyj rezul\tat.
Teoremu 3.2 dovedeno.
OtΩe, neobxidnymy ta dostatnimy umovamy topolohiçno] sprqΩenosti dvox
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
KLASYFIKACIQ TOPOLOHIÇNO SPRQÛENYX AFINNYX VIDOBRAÛEN| 139
afinnyx vidobraΩen\, wo magt\ xoça b po odnij neruxomij toçci, [ neobxidni ta
dostatni umovy topolohiçno] sprqΩenosti ]xnix linijnyx çastyn.
ZauvaΩennq 3.2. Dlq topolohiçno] klasyfikaci] periodyçnyx linijnyx vi-
dobraΩen\ isnugt\ deqki çastkovi rezul\taty, a same znajdeno periody ta
obmeΩennq na matryci, wo vidpovidagt\ linijnym vidobraΩennqm, pry qkyx to-
polohiçna sprqΩenist\ bude oznaçaty linijnu sprqΩenist\ [1 – 8].
Vykorystovugçy cej fakt, zvyçajno, moΩna klasyfikuvaty çastynu tyx
afinnyx vidobraΩen\, wo magt\ xoça b po odnij neruxomij toçci ta linijni ças-
tyny qkyx [ periodyçnymy j zadovol\nqgt\ ti çastkovi vypadky, pro qki v c\omu
zauvaΩenni jßlosq vywe. AdΩe, vykorystovugçy teoremu 3.1, moΩna zrobyty
vysnovok, wo taki afinni vidobraΩennq budut\ topolohiçno sprqΩenymy todi i
til\ky todi, koly dijsni kanoniçni formy ]xnix linijnyx çastyn zbihagt\sq.
1. Kuiper N. H., Robbin J. W. Topological classification of linear endomorphisms // Invent. Math. –
1973. – 19. – P. 83 – 106.
2. Robbin J. W. Topological conjugacy and structural stability for discrete dynamical systems // Bull.
Amer. Math. Soc. – 1972. – 78, # 6. – P. 923 – 952.
3. Cappell S. E., Shaneson J. L. Nonlinear similarity of matrices // Bull. Amer. Math. Soc., New Ser.
– 1979. – 1, # 6. – P. 899 – 902.
4. Cappell S. E., Shaneson J. L. Non-linear similarity // Ann. Math. – 1981. – 113. – P. 315 – 355.
5. Cappell S. E., Shaneson J. L. Nonlinear similarity and differentiability // Communs Pure and Appl.
Math. – 1985. – 38. – P. 697 – 706.
6. Hsiang W. C., Pardon W. When are topologically equivalent orthogonal transformations linearly
equivalent // Invent. Math. – 1982. – 68. – P. 275 – 316.
7. Madsen I., Rothenberg M. Classifying G spheres // Bull. Amer. Math. Soc. New Ser. – 1982. – 7,
# 1. – P. 223 – 226.
8. Schultz R. On the topological classification of linear representations // Topology. – 1977. – 16. –
P. 263 – 269.
9. Palys Û., dy Melu V. Heometryçeskaq teoryq dyyamyçeskyx system: Vvedenye: Per. s anhl.
� M.: Myr, 1986. � 301 s.
10. Cappell S. E., Shaneson J. L. Linear algebra and topology // Bull. Amer. Math. Soc. New Ser. –
1979. – 1, # 4. – P. 685 – 687.
11. Pelgx H. P., Íarkovskyj A. N. Vvedenye v teoryg funkcyonal\n¥x uravnenyj. � Kyev:
Nauk. dumka, 1974. � 119 s.
OderΩano 18.02.08,
pislq doopracgvannq � 16.10.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
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| id | umjimathkievua-article-3009 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:31Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/53/7d940863a2480dbef402db800c2f8053.pdf |
| spelling | umjimathkievua-article-30092020-03-18T19:43:07Z Classification of topologically conjugate affine mappings Класифікація топологічно спряжених афінних відображень Budnyts'ka, T. V. Будницька, Т. В. We consider affine mappings from $ℝ^n$ into $ℝ^n, n ≥ 1$. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from $ℝ$ into $ℝ$ and also for affine mappings from $ℝ^n$ into $ℝ^n, n > 1$, having at least one fixed point and the nonperiodic linear part. Рассматриваются аффинные отображения из $ℝ^n$ в $ℝ^n, n ≥ 1$. Доказана теорема o топологической сопряженности аффинного отображения, имеющего хотя бы одну неподвижную точку, с соответствующим линейным отображением. Получена классификация, с точностью до топологической сопряженности, аффинных отображений из $ℝ$ в $ℝ$, а также тех аффинных отображений из $ℝ^n$ в $ℝ^n, n > 1$, которые имеют хотя бы одну неподвижную точку и чьи линейные части не являются периодическими. Institute of Mathematics, NAS of Ukraine 2009-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3009 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 1 (2009); 134-139 Український математичний журнал; Том 61 № 1 (2009); 134-139 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3009/2767 https://umj.imath.kiev.ua/index.php/umj/article/view/3009/2768 Copyright (c) 2009 Budnyts'ka T. V. |
| spellingShingle | Budnyts'ka, T. V. Будницька, Т. В. Classification of topologically conjugate affine mappings |
| title | Classification of topologically conjugate affine mappings |
| title_alt | Класифікація топологічно спряжених афінних відображень |
| title_full | Classification of topologically conjugate affine mappings |
| title_fullStr | Classification of topologically conjugate affine mappings |
| title_full_unstemmed | Classification of topologically conjugate affine mappings |
| title_short | Classification of topologically conjugate affine mappings |
| title_sort | classification of topologically conjugate affine mappings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3009 |
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