PROCEEDINGS OF THE
AMERICAN MATHEMATICAL
Volume
97, Number
2, June
SOCIETY
1986
A COUNTEREXAMPLE IN DYNAMICAL SYSTEMS
OF THE INTERVAL
HSIN CHU AND XIONG JINCHENG1
ABSTRACT. In [1] it was proved that if the recurrent points of a continuous
map of the unit interval form a closed set, then this map has no periodic point
with period not equal to a power of 2, i.e. this map is of type 2°°. In this paper
we will construct a continuous map of the interval which is of type 2°° and for
which the set of recurrent points is not closed. By such a counterexample
it
may be shown that some of the results announced in [2] are not correct.
First, recall some elementary definitions.
Suppose that /: / —>/ is a continuous map from the interval / = [0,1] into itself.
Let f° be the identity map on /. For any positive integer n define fn = fo /n_1.
A point x G / is said to be periodic if x = fm{x) for some m > 0. In this case
min{m > 0:/m(x) = x} is called the period of x, and the finite set {fm{x):m =
1,2,...} is called the orbit of x.
A point x G / is said to be recurrent if for any neighborhood
for some m > 0.
U of x, fm{x) G U
A subset S of I is said to be invariant if f(S) C S.
A continuous map of the interval is of type 2°° if it has no periodic point with
period not equal to a power of 2.
We will define a map Xoo'-I —* I having no periodic point with period not a
power of 2 such that the set Ä(xoo) of recurrent points is not closed.
1. Let £[a,6],[c,d] denote the linear map from [a, b] to [c,d\ which carries a into
c and b into d, and let £[a,b],[c,d] denote the linear map from [a,b] to [c,d] which
carries a into d and b into c.
For any positive integer k > 3, let
1
A
1
3
1
2
A
A
1
l
and define a map Hk'-I —*I such that
^* l[0,ofc] = £(0,ofcl,[0,c»]i
^k l[o/t,6fc] = MakAl,[ck,4]'
^fc l[6*,Cfc] = f[&*.<:*].[a*.**]>
^k\ck,dk\
= Mc*>d*l.[afci6*]'
***l[dfc,l] = £(d*,l],[6*,l]>
Received by the editors October 15, 1984.
1980 Mathematics Subject Classification.
Primary 58F20, 34C35.
Key words and phrases.
Recurrent point, periodic point, type 2°°.
JOn leave of absence from the Department of Mathematics,
University
of Science and Tech-
nology of China, Hefei, Anhui, People's Republic of China.
©1986
American
0002-9939/86
361
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Mathematical
Society
$1.00 + $.25 per page
HSIN CHU AND XIONG JINCHENG
362
It is easy to see that the map ßk is well defined and continuous.
The following proposition is obvious.
PROPOSITION l.
The set of periodic points of ßk is {0,1, efc}U[a,k,bk]U [cfc,dk],
where efc G [bk, Cfc],in which 0,1, efc are fixed points of ßk and every point in [ofe,6fc]U
[ck,dk] has period 2.
Therefore, ßk is of type 2°°.
2. Suppose fil
—*I is a continuous
map with fixed points 0 and 1. Define a
map ßk* f'-I -* I such that
Vk*f
\l-(ak,bk)=^k
l/-(o_,bfc)
and
Vk * f \{ak,bk}=fl.[c_A] °/° {[«»A.]./It is easy to see that the map ßk * f is well defined and continuous and has fixed
points 0 and 1.
By definition,
ßk* f{[ak,bk}) = {ck,dk] and
ßk * /([Cfc,4]) = [ak,h]-
Hence, we have
(a) in [aj_,f>fc]U [ck,dk] there is no periodic point of ßk * f with odd period.
Obviously, the points 0, 1, and efc are fixed points of ßk* f- If y G (0, ûfc), or if
y G {dk, 1), then y is not a periodic point of ßk * f because there is some m > 0 such
that (ßk * f)m(y) G [ofc,dfc],and [ofc,dfc]is an invariant subset of ßk * f ■ Similarly,
if y G (bk, Ck) and y ^ efc, then ?/ is not a periodic point of ßk * f because there is
some m > 0 such that (ßk * f)m(y) G [afc,6fc]U [ck,dk] and [ak,bk] U [cfc,dfc]is an
invariant subset of ßk * f ■ Thus,
(b) in [0, dfc) U (6fc,Cfc)U (dfc, 1] there are no periodic points of ßk* f other than
the fixed points 0, 1 and efc.
In addition,
= M* I[Cfc.rffc]
° ^.[c*.«í*] ° f° ^[ak,bk],I
= ^[cfcldfc],[afc,6fc] ° C/,[ck,(ifc] ° / ° C[a„,6„],/
= í/,[afc,6fc] ° f ° tlak,bk],I-
From this, it follows that
(c) for any m > 1,
Uk * /)2m
By statements
\[ak,bk]= ^,[a„,6fc] « /m o €l«»AJ,/«
(a), (b), and (c) one can prove the following proposition
immedi-
ately.
PROPOSITION 2. (1) // there is a periodic point of ßk * f with period n > 1,
then n is even and there is a periodic point of f with period n/2.
Therefore, if f is of type 2°°, then ßk * f is of type 2°°.
(2) If x is a periodic point of f with period rn > 0, then y = £¡ tak,bk}(x) îS a
periodic point of ßk* f with period 2m; furthermore, if for some uG I
fl(x)>u
for 1 = 1,2,..., m - 1,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
DYNAMICAL SYSTEMS OF THE INTERVAL
363
then
(ßk * f)l(y) > ù,[ak,b„](«) for I = 1,2,...,2m -1.
PROPOSITION 3. Suppose that f,f':I
points 0 and 1.
—►/ are continuous maps with fixed
(1) // i/iere is some M > 0 suc/i í/iaí
|/(z)-/'(x)|
<M
for all x El,
then
1/4 * /(V) - Mfe* /'(y)| < M/2fc /or a// y G /.
(2) ///
and /' coincide on I — (u,v),
where [u,v] is a subinterval of I, then
ßk* f and ßk * f coincide on
/-(&,[•*
Ak](tt)>(/,[ak Ab](v))-
PROOF. (1) If y G I-(ak,bk), then ßk*f (y)-ßk*f'(y)
= 0, and if y G [ak,bk},
then
ßk * f(y) -ßk* f'(y) = ti,[ck,dh](f(Ziak,bk]Av)))
- £l,[ck,dk](f'(£{ak,bk]j(y)))
= &,[cfc,dfc](/(z))
- tl,[ck,dk](f'(x))
= (dk - Ck)(f(x) - f{x))
= ¿(/(*)-/'(*)),
where x = í[at,bt],/(y)-
Hence for any y el,
\ß*f(y)-ßk*f'(y)\<M/2k.
(2) By the definition, ßk* f and /¿fc* /' coincide on _"- (at, 6fc). Since / and /'
coincide on I — (u, v), ßk * f and ßk * f coincide on
Éh.^l./ÍJ-M))
= &,[•*••_](*)
- £/,[«*
,6_]
((«,«))
= he A]
- (6,[ofc,bfc](w),6,[afc,.„](w))-
Therefore, ßk* f and /-fc * /' coincide on
I - (Çl,[ak,bk](u)rÇl,[ak,bk](V))-
3. Let
«3=23,
1_
¡3
i
04 ~ 23 + 2 ' 24'
1
11
Q5~23+2'24+2'4'25'
J_
afc-23
1 J_
+ 2'24
13
1
1 3 J_
+ 2'4'25+'"
13
+ 2'4'"
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
2fc~3-1
2fc"3
1
'2fc'
HSIN CHU AND XIONG JINCHENG
364
and let
_L_J_
A = i - ^23
-2 A
24'
^5 ~
23
Pk -
^_1_3__13^
23 2 ' ¥
Obviously,
ßk-l
2 ' 24
2 ' 4 ' 25 '
2 ' 4 ' 25
13
2*4
'"
2fc~3- 1 3^
2fc"3 '2fc'
Q3 < Q4 < a5 < ■• ■ < Ofc_i < a.k < ■■■ and ß3 > ß4 > ß5 > ■■■ >
>&>•••.
Since both sequences 03,04,...
and ßz,ßi,..are monotone and bounded, their
limits exist. Let a = limfc-^oo Qfc and ß = linifc-^-o ßk- By a simple calculation, we
have that
13
ßk
2k~2 - 1
Qfc ~ 2 ' 4 "
2k~2
'
Thus,
/3-o = fclim(Ä-afc) = n(l-i)>0
¿=i ^
'
(see [4]). Hence,
(*)
Q3 < C-4 < • • • < Ctfc-1 < «fc < • " < a < ß < • • • < ßk < ßk-l
< ■■■ < ßi < ß3-
By the definition of the notation, £_,[,_,_],£_,[■_.,-]
(x) = u + (v —u)x for any x e I.
Therefore,
£-,[-.3,-3] ° í/,[04,fc«] ° • • • ° É-,[a*_i,&*_._](<»fc)
= a3 + (63 - a3)£/i[a4i_4]
o • ■■o C/.iafc-Lbfc-tiíafc)
= 03 + (03 - 03)04 H-r-
111
13
~ ¥ + 2' ¥ + '" + 2' 4
(63 - a3)(64 - 04) ■■• (-fc_i - afc_i)afc
2k~3 - 1 1
2fc-3 ' ¥
= Qfc-
In brief,
(**)l
ak = Ù,[a3,b3] ° 6,(04,64] ° ■• • ° Éï,Ia__i,6»_i)(ak),
and by a similar calculation,
(**)2
ßk = C/jag,^]
o(,,|aj4]
o-.-o^j-^^^.j^fc).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
DYNAMICAL SYSTEMS OF THE INTERVAL
365
4. Now, we define
X3 = M3,
X4 = ß3 *Pi,
Xh = ß3 * (ß4 * ßb),
Xk = ß3 * (ßi * (• ■■* (ßk-1
Proposition
*ßk)-
■))-,
4. (l) \k is of type 2°°.
(2) Xfc+i \i-{oik,ßk)=
Therefore, ifl>k,
Xk |/_(afcÄ)-
then
*k
\i-(ak,ßk)=Xl
\l-(ak,ßk)
■
(3) Ofc is a periodic point of \k with period 2k~2, and
(Xk)m(oLk)>ßk
for m = 1,2,...,2k~2-l.
Therefore, if I > k, then ctk is a periodic point of xi with period 2h~2, and
(Xi)m(ctk)>ßk
for m = 1,2,...,2k~2-l.
(4) For any x e I,
|xfc+1(*)-Xfc(*)|<2-<fe+3>(*-2)/2.
PROOF. (1) Propositions
1 and 2(1) imply the statement
(1) of this proposition.
(2) By the definition, ßk * ßk+i and ßk coincide on I —(ak, bk). By Proposition
3(2), ßk-i * (ßk * ßK+i) and ßk-i * ßk coincide on
I - (£/,[ofc__,&fc_.](ak),6,(afc_i,4*__](&k))-
Applying Proposition
I -
(£.,[o3,63]
3(2) repeatedly, we have that Xfc+i and Xfccoincide on
° £-,[0.4,64]
° ••• ° £/-[afc_1,6fc_1](afc),
£.,[a3,63]
° £/,[o4,64] ° ■• ■° tl,[ak-.ubk-i](h)),
which is I —(ctk,ßk) by (**)i and (**)2(3) üfc is a periodic point of ßk with period 2 and ßk(a-k) — c-k > bk- By
Proposition 2(2), í/jo,,.^-^^/-)
is a periodic point of ßk-i * ßk with period 4,
and
(ßk-i
*MK)m(£/,[o^1,6fc_1](afc)) > C/.lo^A^K&fc)
for m = 1,2,3.
Applying Proposition 2(2) repeatedly, we have at = £_,[a3,b3] ° £.,[04,64] o • • • o
ù,[ak-ubk-i](ak)
is a periodic point of Xfc with a period of 2k~2, and
(Xk)m(0Ck)
> £/,[o3,63]
° £.,[04,64]
° • ' ' ° il^k^i^.^h)
form=l,2,...,2fc-2-l.
(4) By the definition, for any x e I
\ßk * ßk+i(x)
- ßk(x)\ < 2~k.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
= ßk
366
HSIN CHU AND XIONG JINCHENG
By Proposition 3(1), for any x e I
|/ifc-i * (ßk * ßk+i)(x)
Applying Proposition
\Xk+i(x)
- ßk-i
* ßk(x)\
< 2_(fc~1) ■2~k.
3(1) repeatedly, we have that for any x e I
- Xk(x)\ = \ß3 * (/Í4 * (••• * (ßk * ßk+i)
- ß3 * (ßk * (■ ■■* (ßk-1
< 2-3-2~4
• ■-))(x)
*ßk)-
-))(x)\
■■■2~k
_ 2-(fc+3)(fc-2)/2
5. By Proposition 4(4) the sequence of the maps X3, X4>• • • is uniformly convergent, and its limit map, denoted by Xco, is continuous. By a theorem of Block (see
Main Theorem in [3]), Xoo is of type 2°° because all Xfc's are of type 2°°.
Now, we will state and prove our main conclusion as follows.
THEOREM. The set of recurrent points of the continuous map x_o: I —*I> which
is of type 2°°, is not closed.
PROOF. By Proposition
4(2) and the definition of Xoo it is easy to see that
*°°
\l-(ak,ßk)=Xk
\l-(ak,ßk)
■
Hence, by Proposition 4(3), Ofcis periodic of x°° with period 2fc~2, and
xS(Qfc) > ßk > ß
Consider the accumulation
any positive integer m > 0
form=l,2,...,2fe-2-l.
point a — limfc_0o ctk of periodic points of Xoo- For
*£(<_) = lim XZ(ak).
k—»oo
Thus, xS > ß because xS(öm+2)
> ß, xS. («m+a) > A • • •• Since a < ß (see (*)),
a is not a recurrent point of Xoo• This shows that the set of recurrent
points is not
closed.
References
1. Xiong Jincheng, The periods
whose recurrent points form
2. A. M. Bloh, The asymptotic
of periodic points of continuous
self-maps of the interval
a closed set, J. China Univ. Sei. Tech. 13 (1983), 134-135.
behaviour of one-dimensional
dynamical
systems, Uspekhi
Mat. Nauk 37 (1982), 175-176.
3. L. Block, Stability
of periodic
orbits in the theorem
of Sarkovski,
Proc. Amer. Math. Soc.
81 (1981), 335-336.
4. A. Weil, Two lectures
on number
theory,
past
and present,
Enseign.
Math.
20 (1974),
87-110.
Department
land
of Mathematics,
University
of Maryland,
College
20742
International
centre
for Theoretical
Physics, Trieste,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Italy
Park, Mary-