A N N A L E S SO C IE T A T IS M A TH EM A T IC A E PO LO N A E Series I : CO M M EN TA TIO N ES M A TH EM A TIC A E X I X (1977) R O C Z N IK I P O L SK IE G O TO W A RZY STW A M ATEM ATYCZNEGO
Séria I : P R A C E M A TEM A TY CZ N E X I X (1977)
Ed w ard S4SIADA and Paw ed Ja r e k (Torun)
On the conjugacy of homeomorphisms of unit interval
Introduction. Homeomorphisms xx and r2 of unit interval I = [0,1]
are said to be conjugate, if there exists a homeomorphism or of I such that axx = r 2cr (write rx ry r2 or, shortly, xx ~ r2).
Conjugacy is an equivalence relation on the group H (I) of all homeomorphisms of I.
The purpose of this paper is to describe the equivalence classes under this relation, that is to formulate necessary and. sufficient con
ditions for homeomorphism r, and r2 to be conjugate.
It is well known that every homeomorphism of I is a monotonie function, thus the group H (I) may be decomposed into disjoint subsets H (I) ^ H x u H 2, where H x is the set of all increasing, and H 2 the set of all decreasing homeomorphisms.
Recall some elementary properties:
(1) E x is a normal subgroup of E (I), (2) r *€H x for any XeE(I),
(3) rx ~ r2 and хх* Е { implies r%^E i for i = 1 , 2 . Let F (r) be the set of all fixpoints of r, that is
F(x) = {æel: x{æ) ==&},
(4) F(x) is a closed subset of I, containing 0 and 1 iff x c E x and card F (r) = 1 iff x e E 2.
(5) If reH x, then for any x e l the sequence {rn(a»)}n_1A... is conver
gent to a fixpoint of r. ;
(6) If x e E x, then F (xn) = F(x) for any n Ф 0.
(7) If xeff2, then F { т2и+1) = F(x) for any n and F ix 2,4) = F (x 2) for any n Ф 0.
Let Х€Лг. Then x eF {x 2) iff x(x)eF{т2). 0
(8)
I. Conjugacy of increasing homeomorphisms. Let F be a closed subset of I, containing 0 and 1. The open set I \ F is a (finit or countable) sum of pairwise disjoint open intervals (av,b v) (veN(F)). Denote L {F )
= and B (F ) = {bP}veN(F). Let lF : L ( F ) ^ { 0 , 1} be an arbi
trary 0-1-value function on L (F ). Then a function rF : B (F )-+ {0 ,1 } may be defined by formula tf(Pv) — 1 lF (aP).
Let F x and F 2 be closed subsets of I, containing 0 and 1 and let lF : L(jFT1)-?-{0,1} and Zpy L ( F 2)~>{0, 1} be 0-1-value functions. The pairs (F x, lFi) and (F 2, lF ) are said to be equivalent, if there exists a homeomorphism cr of I such, that
1° a {F x) = F 2;
2° lFl(ap) = l Fz(o{aP)) {v e N iF J) if аеН г (i.e., if a {L {F x)) = L {F 2)), lFx{av) = rF2(a(aP)) (v eF fëi)) if aeH 2 (i.e., if a{L{Fx)} = B ( F 2)).
If these conditions are fulfilled, then write (F x, lFi) ~ (F 2, lF ) or shortly, {F x, lFl) ~ (F 2, lFi).
Relation ^ defined above is of course an equivalence relation on the set of all pairs (F , I f ).
Now, let Tcjffi and let I \ F ( r ) = ( J (ap,bp). We shall write L(x) veN(r)
(resp. E(r)) instead of L[F(x)) (resp. B(F{x))).
Whence L(x) = K } veiv(T), B(x) = {&,}„ciV(T).
Define lr: L (r)-> {0 ,1 ) as follows:
lr{aP) J 0 if x(x) < x for all xe(av, bp), l l if x < x ( x ) for all хе(ал,Ър).
Function lr is well defined, since r has no fixpoints in (ap, bv).
Observe, that
(9) lr(av) — 0 iff lim tn(x) — aP and lim x~n(x) = bp
fl—XX) n —>oo
for any xe(aP, bp), lx{ap) = 1 iff lim rw(a?) = bP and И т т -И,(а?) = av
0 0 f l- V O O
for any X€(ap, bp).
Th e o r e m 1. Let x 1,x 2cH1. Then xx~ x2 iff ( F (xx), lX])~ (F(x2), lxJ . P ro of. 1. Assume x1~ x 2. Take X€F{xx). Then we have a{x)
= a{xx{x)) — х2{а(х)), whence o(x)eF(x2). Similarly if y eP (r2), then a~1(y)eF(x1). Thus a{F(xx)) = F (x 2). Suppose aeH x. Then ^(^(r!))
= L {r 2). Take aPeL(x1) and suppose lt (ap) = 0. Then xx(x) < x for any xe(aP,b v), whence x2(o(x)) = a{xx(x)) < a{x), that is r2( y ) < y for any y €(a(ap), <j(bp)). Thus l4 (a{ap)) = 0 .
It may be similarly proved, that Ц(а„) = 1 implies lX2{a{av)) — 1.
Therefore (F(xx), ZTJ ~ (F {r2), ZTj .
Conjugacy o f homeomorpMsms 367
The proof is similar provided aeH 2.
2. Assume (F(xx), Ц) ~ (F (t2), ZTJ.
We shall construct a homeomorphism q oî I such, that qxx — x2q. Suppose a e S x (the proof is similar provided aeH 2). Let I \ F ( x x) —
= U iaviK)- Then I \ F { x 2) = U (a {av)i First we shall con
vex^ T j) veN(t j)
struct (for any V€N(xx)) homeomorphism
qv: (av,b v)^ [a {a v), a{bv))
such, that qvxx(x) = x2gv(x) for any xe(av, bv). Suppose lTl(av) = 1 (the construction is similar if lr {av) = 0). Fixing a point xve(av, bv) we have in view of (9):
(10) (av,b v) = ( J [т?(а?,), + 0O t“+1(®,)],
n = — o o
+ o o
(o{av), a{bv)) = ( J [r£(<r(a>,)), xl+l(a{xv))\.
n = — oo
Let [xv, xx(xv)]-> [a(xv), x2(o(xv))\ be a homeomorphism such that Xv(xv) = a(xv) (Av may be for example a linear mapping). Put
qv(x) = x2l vx fn(x) if X€[xy (xv), Xi+1(xv)~\.
It is easily seen, that gv has the desired properties.
Now, define q: I->1 as follows:
q(x) f o(x) l QV{X)
if X€F(xx),
if xe(av, bv) {veN{ X X) ) . It is evident that q is a homeomorphism of I and qxx = x2q.
The proof is completed.
Now we shall formulate some corollaries.
Corollab y 1. Given any closed subset F of I containing 0 and 1 and any 0-1-value function lF : L (F )-> {0 ,1 } there exists a homeomorphism xeHx such that F = F(x) and lF = lT.
P roof. The statement is obvious.
Co r o l l a r y 2. Let xx, x2eHx and F (x x) = F (x 2) = { 0 , 1 } . Then xx ~ r 2.
P roof. It is easily seen, that [F{xx), У ~ [F(x2) ,l T2], the state
ment follows thus from Theorem 1.
Co r o lla r y 3. Let xeHx. Then x ~ xn for any n ^ l .
P ro of. We have F(x) = F {x n) as stated in (6). Furthermore, if x < x(x) (resp. x(x) < x), then x < xn(x) (resp. xn(x) < x) and therefore lt = l . Thus the statement follows again from Theorem 1.
Co r o l l a r y 4. For any t e R 1 and any n Ф 0 there exists aeH x such that r = an.
P roof. Assume n > 1. Then т ~ tn, whence there exists qcH (I) such that Qt = rn q. Therefore r = q^ t71 q = (f>_1 vq)71. Put a = q~1xq. The case n < 0 follows therefore by the obvious equality an = (o,“ 1)"re.
II. Conjugacy of decreasing homeomorphisms. Let F be a closed subset of I containing 0 and 1 and let x0<=F. F will be called œ0-symmetric if there exists a homeomorphism a of I such, that:
1° a is decreasing, 2° a(x0) = xQ, 3° a (F) = F ,
4° or2(a?) = x for any xeF.
E e m ark . If a has properties 1°, 2° and 3°, then we may define another homeomorphism a with properties 1°, 2°, 3° and 4° as follows:
\a{x) if x e [0 .x 0], a(x) = 1
[or"1 (x) if #e[a?0, l ] .
Let F be a;0-symmetric, F 0 = F n [0, x0] and [0, x0] \ F 0 = U (av, bv) and let a establishes the avsymmetry of F . ™n(F)
Then
F = F 0kja (F 0), F 0n a ( F 0) = {xQ}, [x0, l]\< r(F 0) = I J (a(bv), a(av)),
reN(F)
^ iF ) = {ог(&у)}уеЛг(^> >
R (F) = {bv}vejy(F) v {o(av)}veN(F) .
In this case the 0-1-value function lF : L (F )-> {0 ,1 } is said to be sym
metric, if lF (av) = rF {o{av)) for any veN(F).
Let F { be ^-symmetric and lF.: L (F {) ^ { 0 ,1 } symmetric 0-1-value functions (i = 1,2).
The triples (F 1,x 1, l Fi) and (F 2,x2,If ) are called equivalent, if there exists a homeomorphism q of I such that:
1° q(F x) = F 2,
2° q{xx) = x 2,
3° lFl{%) = ^ 2(еЮ ) if q[L {F x)) = L { F 2), that is, if qcH 1, Ц К ) = » ’r , ( e M if e(£(#i)) = ЩЯ1*), i-e., if q*H 2.
If such q exists, then write (Fx, xx, lFi) ~ (F 2, x2, lF ) or, shortly, (Fx, xx, lFl) ~ (F 2, x2 , Ц ).
Conjugacy o f homeomorphisms 369
Relation ^ defined above is an equivalence relation on the set of all triples ( F ,x 0, lF).
Eow, let reJT2 and let xx be the unique fixpoint of r. It is easy to see that F ( r 2) is «^-symmetric (indeed, r establishes the symmetry, cf.
(8)) and lr2 is symmetric too.
Th e o r e m 2. Let rx, x 2e H 2. Then xx ^ r 2 iff (F(xf), xT , 12) ^
~ (-F(rf), * Г2, О)-
P roof. 1. Assume r1r^’ x2. Then x2x~ x l and evidently
2. Assume
(F{rl),a>Ti, l z ) ~ (F(rt), æT2
№ ) , ,*2 (F(rl),œ T2
We shall construct homeomorphism q of I such that qxx= r2 q. Assume a is increasing (the proof is similar provided a is decreasing).
Let F x = F(rl) n [0 , <ctl], F 2 = F { r 22)n [0, xX2] and [ 0 ,x Xl\ \ F x = U ((*v,K). Then a ( F x) = F z, [0, xTJ \ F 2 = U ( * ( « ,) , <r(h)).
veN( t j ) veiV (ri)
We shall construct (for any veN(xx)) homeomorphism
qv: (av, bv)u{xx{hv), гг(а9))-+(о(а,), <r(&„))u (r2(cr (&„)), x2(o{avj))
such that Qvxx{æ) = x2qv{x) for any xe(av, bv)u(xx(bv), xx(av)). Suppose 12{av) = 1 (the construction is similar if 1 2(av) = 0). Fixing a point
ri Ti
xve(av,b v), we have in view of (9):
+ 00
{ч(Ю , *>(».))
+oo
= U (*»)],
72, a s — OO
+oo
((т(а„), or(6,)) = U [rf(or(a!j), t f +2(<r(®,))], 72= —00
+ 00
(r3((7(6v)), r2((T(^))) = (J [rf+1(<r(av)), x f ^(a?,))]
Let Av: [xv, x?(av)]-*[<r(®,), т2((т(а;,))] be a homeomorphism such that Xv(xv) = a(xv) (A„ may be for example linear mapping).
Put
qv{og) I x f Xvx f2n{x) if
\xln~ 4 vx f2n+l {x) if
X€[xln (xv), x\n+2(xv)], x * \ x f + l{xv), rf-^a?,)].
It is easy to check that qv has the desired properties.
Now define q: I - > I as follows:
q(x)
f a{œ) if x e F x, x2a t f l (x) if x e r^ F f),
Qv{æ) - if xe{av, b ju f a f i,) , тх(а,)) (геЖ Ы ), It is evident that q is a homeomorphism of I and qtx — t2q. The proof is completed.
E em ark . Observe that homeomorphism q constructed above is increasing. But if a is decreasing, then the construction, described in proof of Theorem 2 leeds again to a decreasing homeomorphism q such that t2. However, in this case we may take homeomorphism a = r2a instead of a. It is easy to see that a establishes again equivalence between the triples (F (т \),х г , l 2) and {F(t^),xT , l 2) and a is increasing.
V 1 T1 V 2 T2
So we have proved:
Co r o l l a r y 5. I f rx, т2е Я 2 and rx ~ r 2, then there exists д е Н г such that r1r^J r2.
Co r o l l a r y 6. Given any triple (F , xQ, lF) composed of x0-symmetric closed subset F of I containing 0 and 1 and symmetric 0-1-value function
lF: L (F )-^ {0 , 1}, there exists t e H 2 such that x0 = xz, F — F ( r2), lF = lT . P roof. Let [0 ,x o] \ F = U (av,b v) and let a establishes a?0-sym-
vcN(F)
metry of F . Let A„: [a(bv), a(av)]-^[av, bv] be linear mapping such that Xv{a{av)) = av. A„ is a decreasing function, thus A'(a?) < 0. Let qv: [av, &„]->•
->[cr(6J, <y(av)] be a homeomorphism such that Qv(av) = a(av) and q'{x)
< 0 if lF (av) = 0, q'{x) > 0 if lF
r : I - ^ I as follows:
K ) = 1 for any xe[av, bv] . Now define
a(x) if x * F , x = • Qv(x) if § W 1_1 « * ъI---1
A(®) if xe[a(bv), <r(a,)];
r is evidently a decreasing homeomorphism of I, r(x0) = x0 and F F (t2).
If xe(av,b v), then we have r 2(x) — Avqv{x) and {r2(x))" = K (qv(x))q'v'(x).
Whence, if lF(av) = 0, then (r2(a?)),/ > 0 for any xe(av, bv) and therefore r 2(x) < x for xe(av, bv), that is lz2(av) = 0. If lF {av) = 1, then (r2(a?))" < 0 and r 2(x) > x for any xe(aP, bv), whence lr2{av) = 1. Now it follows from inequalities r 2(x) < x resp. r 2(x) > x for xe(av, bv), that F ( r 2) <= F , and thus F ( r 2) = F.
Co r o l l a r y 7. Let reH 2. Then t ~ t2w_1 for any n ^ l .
Conjugacy o f homéomorphisme 371
P roof. We have r 2eHx, thus, in view of Corollary 3, F (т2) = F ( ^ 2n~l)) and 12 = ?T2(2n-i)> Furthermore xx = x^n_v Thus the conjugacy x ^ r2”-1 follows from Theorem 2.
Co ro l l a ry 8 . For any reH 2 and any integer n there exists oeH2
such that r = cr2n-1.
P roof. Is similar to that of Corollary 4.
We shall finish our considerations with an example of homeomor- phisms xx, r2eH2 such that x\ ~ %\ but xx op x2.
Let F = (0, f , l } u { l jn, n = 3 , 4 , . . . } u { l —1 /n, n = 3 , 4 , . . . } and x0 = F is closed a?0-symmetric subset of I containing 0 and 1. Define two 0-1-value functions l{: L (F )-+ {0,1 } (г = 1 , 2 ) .
Observe that it sufficies to define li on left borders of components of I \ F contained in [0, •£], that is on the points an = 1/n (n = 3, 4, ...).
Put
0 if n II
1 if n = 2&+1;
1 if n = 2 fc, 0 if n = 2ft+ 1 .
There exists, by Corollary 6, homeomorphisms xx, x2e H 2 such that
*rt = h ЩА) = Р , 1 2 = 1 { (i = 1 , 2 ) .
We shall show, that xx r2. Assume, in contrary, that xx ~ r2 and let homeomorphism a establishes the conjugacy, that is oxx = x2a. In view of Corollary 5, a may be chosen increasing.
Now, it must be <r(|) = ^, but this is impossible, since 12{\) = 1
and Z 2(з) = 0. Tl
*2
We shall show, that x2x *^2 *
Let q be an increasing homeomorphism of I such that q (1 jn) = 1 J(n -f 1), e ( l — l/(w+l)) = 1 — ljn (n = 2, 3, ...). Then q(F) = F and lx = l 2Q, whence, in view of Theorem 1, x\ ~ r 2.
