BULL. AUSTRAL. MATH. SOC.
5 8F20
VOL.
(IOAIO,
31 ( 1 9 8 5 ) , 8 9 - 1 0 3 .
26AI8)
THE MINIMAL NUMBER OF PERIODIC ORBITS
OF PERIODS GUARANTEED IN SHARKOVSKII'S THEOREM
BAU-SEN DU
Let
f(x)
be a continuous function from a compact real interval
into i t s e l f with a periodic orbit of minimal period
is not an integral power of
for every positive integer
2 .
n
m , where m
Then, by Sharkovskii's theorem,
with
m -*• n
in the Sharkovskii
ordering defined below, a lower bound on the number of periodic
orbits of
f(x)
with minimal period
improve this lower bound from
1
n
is
1 .
Could we
to some larger number?
In this
paper, we give a complete answer to this question.
1.
Let
a;
in
I
J
be a compact real interval and l e t
and any positive integer
iterate of
(under
/ ).
x~
under
If
/
a periodic point of
(under
Introduction
/
and call
[xA = x^
f
•(/C(x0) | k > 0 \
x
the orbit of
/
xQ
m , we call
and of the orbit (under
m , must
For any
denote the feth
and call the cardinality of the orbit of
has a periodic orbit of a period
n # m1
f (xn)
for some positive integer
/ ) the minimal period of
of periods
k , we l e t
f € C (T, J) .
x
x
/ ).
If
/
also have periodic orbits
In 196U, Sharkovskii [17] (see [ J ] , [ 3 ] , [ 7 ] , [ 9 ] ,
Received 23 August 198U.
Copyright Clearance Centre, Inc. Serial-fee code:
$A2.00 + 0.00.
000U-9727/81t
89
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90
Bau-Sen Du
[72],
[73], also) had given a complete answer to this question.
Arrange
the positive integers according as the following new order (called
Sharkovskii ordering):
3
A
A
A
7
A ...
A 2-5
A
2*7
A ...
2 * - 3 A 2 W «5
A
2n«7
A ...
A
22
2*3
A
A
...
5
A 2
3
A
Sharkovskii's theorem says that any function
orbit of minimal period
minimal period
2
A
1
.
f € C (I, I)
with a periodic
m , must also have at least one periodic orbit of
n precisely when
m A n in the above Sharkovskii
n with
m A n , the
ordering.
Therefore, for every positive integer
number
is a lower bound on the number of distinct periodic orbits of f
1
with minimal period
n . One question arises naturally:
this lower bound from
could we improve
1 to some larger number?
In 1976 Bowen and Franks [2] showed, among other things, that if
/ € C (I, I)
m >1
has a periodic orbit of minimal period
is odd, then there is a number
for all integers
k 2: M
M
(independent of f ) such that,
, / has at least
periodic orbits of minimal period
n = 2 m , where
(2 'm)/[2<lt) distinct
2k.
In 1979, Jonker [S] also obtained a similar result on a class of
unimodal maps.
If 0 is an interior point of J , let
collection of all
f (. C (J, I)
S
G
denote the
which has either one maximum or one
minimum point at c , and is strictly monotone on each component of
I - {a} with
m, n
/( dl) c 81" . Jonker showed, among other things, that if
are any two odd integers with
1 < m < n , and if f € S
has a
c
periodic orbit o f minimal period
/
must also have at least
period
2
2m, where
k i 0 is any integer, then
distinct periodic orbits of minimal
2n.
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Per i od i c orb i t s
91
In [6], a result along this line is also obtained. However, that
result is only a partial one. In this paper we give a complete answer to
that question.
In Section 2 we state our main results (Theorems 1, 2, and 3). In
Section 3 we describe the method used to prove them. This method is the
same as that used in [5] and [6]. The proofs of Theorems 1 and 2 will
appear in Sections h and 5. Theorem 3 then follows easily from Theorems 1
and 2.
2.
Let
Statement of main results
<{)(m) be an integer-valued function defined on the set of a l l
positive i n t e g e r s .
prime numbers,
\ k2
k
m=p p
... p
If
r
and
fe.'s
, where the p . ' s
are d i s t i n c t
are positive i n t e g e r s , we define
and
I
*3
4 > ( m / ( p . p . p . ) ) + . . . + ( - l ) r t ( » / ( p p . . . p ))
1 2 3
1 2
r
,
where the summation
15 i
£
is taken over all i , •£„, ..., i . with
3
.
i <i<...<i
1 2
Q
< ... < i . S r .
I f , when c o n s i d e r e d a s a s e q u e n c e , < <(>(m) >
< i
-L
^
0
is the Lucas sequence, that is if
<i>(m+2) = (j>(m+l) + 4>(m)
simplicity, we denote
<J>(l) = 1 ,
<J>(2) = 3 , and
for all positive integers
H-m, <j>) as
and if, for every positive integer
tAm)
m ,
.
<j)(m)
m , then, for
Note that if
/ € C (I,
I)
is the number of distinct
solutions of the equation f (a;) = x , then $(m, (J>) is , by the standard
inclusion-exclusion argument, the number of periodic points of / with
minimal period m . Now we can state the following theorem.
THEOREM 1.
Let
f : [ l , 3] •+ [ l , 3] be defined
by
f(x)
= -2x + 5
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Bau-Sen Du
92
if
1 5 x 2 2 and f(x) = x - l £ / 2 < x < 3 .
(a)
for every positive
integer
distinot
of the equation
solutions
sequence
(b)
(c)
integer
periodic
the sequence
and
a
hold:
is the number of
j (x) = x , then the
( a ) is the Lucas sequence;
for every positive
distinct
m , if
Iften the following
m,
f
has exactly
of minimal period
orbits
( $Am)/m}
is strictly
§Am)/m
m;
increasing
for m > 6
l i m [$ (m+1)/(m+1)] / [$Am)/m] = (l+\/5)/2 ..
n"*>
n-"*>
Fix any integer
n > 1 and l e t
Q = { ( 1 , n+1)} u {(m, 2n+2-m) | 2 5 m < «}
5 m S 2n}
For a l l integers
i , j , and k , with
define £>,
k
recursively as follows:
1 , if
b
.
.
±,i-,3,n
1 < i , j S 2n and k 2; 1 , we
(i, j) €
= •
0 , otherwise,
and
k
if
1 5 j < n-1 ,
k,i,n+X,n
if
j =n ,
if
3 =n +1 ,
if
n+2 5 j 5 2n .
ki2n+ljn
k,i,n,n
b
k,i,l,n
'
k,i,2n+2-j,n '
We also define
e,
by l e t t i n g
2n
C
k,n
Note that these sequences
k,n+l,n,n
(b, . .
) and (c,
\ have the following six
properties. Some of these will be used later in the proofs of our main
results. (Recall that n > 1 is fixed.)
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Per i od i c orb i t s
(i)
The sequence (b,
k > 2 , we have
93
) i s i n c r e a s i n g , and for a l l integers
Z>, ,
> i>,
and 2>, ., . .
> b,
.
k,l,n,n
k,n+l,n,n
k,l,i+l,n
k,l^,n
for a l l
1 5 i 2 n-1 .
(ii)
The sequences <ifc>1>J->n> , 1 S j < n , and <* fc>n+1 , n>B ) =an
also be obtained by the following recursive formulas:
b
2,lJ,n
l,n+l,n,n
=1 '
1 =^
»•
~ 2,n+l,n,n
'
For i = 1 or n + 1 , and fe i 1 ,
(iii)
k+2,i,n,n ~ k,i,l,n+
k+l,i,n,n
'
k+2,i,j,n
k,i,j+l,n
'
~ k,i,l,n
For every positive integer
k , a,
o
_
can also be
obtained by the following formulas:
~ fe+2w-2,n+l,n,n
+
n
A
k+2n-2j ,1 ,j ,n
3 ~^-
k+2n-2,n+l,n,n
n
.^
k,l,n,n
The first identity also holds for a l l integers
provided we define
(iv)
(v)
e
, n
K
fc,
j-l- J<7 >
For a l l integers
For a l l integers
= 2a
(vi)
2k+l,n+l
n
k with
k,l,n+l-i,n
-2n+3 5 k 5 0
= 0 for a l l -2n+3 SfeS 0 and 1 5 j < n .
k with
k with
fe+1
1 5 k S 2n , e 9 ,
= Z~
- 1.
n+1 < k < 3« ,
~1 '
Since, for every positive integer
2n
b
I. ~ )
k,l,n,n
=b
k-l,X,n,n
+
k 2 2«+l ,
i
(
-?P ~
l )h
k-i,\,n,n
'
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94
Bau-Sen Du
there exist
i>,
K,±sn,n
=
Zn + 1
nonzero constants
k
£
a .x . for all positive integers
•
t h e polynomial
is the set of all zeros (including complex zeros) of
a;
- 2x
- 1 .
For a l l positive integers
Yt
k , where
Q Q
\x. | 1 2 J S 2n+l}
3
<>
j (fe) = c,
cx.'s such that
3
and l e t
k, m, n , with
n > 1 , we l e t
$ (m) = $(m, (> ) > where
K. ^Yl
YL
$ is defined as above.
Yl'
Now we can state the following theorem.
THEOREM 2.
For every integer
fn
n > 1 , let
: [ 1 , 2M+1] -»• [ 1 , 2n+l]
2?e t?je continuous function with the following six
(1)
/n(l) = n + 1 ,
(2)
f n (2) = 2n + 1 ,
(3)
f n (« + l) = « + 2 ;
(U)
/ n (n+2) = n ,
(5)
fM(2n+l) = 1 , and
(6)
_f
i s linear on each component of the complement of the set
{2, n+1, n+2} in
Then the following
(a)
(c)
c,
integer
distinct
for every positive
distinot
[ l , 2n+l] .
hold:
for every positive
exactly
(b)
properties:
periodic
k , the equation
positive
n
=x
has
solutions;
integer
orbits
m3
f (x)
has exactly
of minimal period
lim (log[_4> (m)/m])/m = X , where
m**>
j(x)
n
X
$ (m)/m
m;
is the
(unique)
(and the largest in absolute value) zero of the
,
. .,
2n+l
„ 2n-l
polynomial
x
- 2x
- 1 .
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Period i c orb i t s
95
From Theorems 1 and 2 above and Theorem 2 of [72, p . 2l»3], we e a s i l y
obtain the following r e s u l t .
THEOREM 3.
Assume that
f € C°(I, I)
has a periodic orbit of
minimal period s = 2 (2n+l) , where n > 1 and fe 2 0 , and no periodic
orbits of minimal period r with r A s in the Sharkovskii ordering.
Then for every positive integer
t with s A t in the Sharkovskii
ordering,
f
$n{t/2k)/[t/2
has at least
orbits of minimal period
)
(sharp) distinct
periodic
t .
REMARK I. We call attention to the fact that there exist continuous
functions from I into J with exactly one periodic orbit of minimal
period
2
for every positive integer
i
(and two fixed points), but no
other periods (see [JO]).
REMARK 2. With the help of Theorem 2 of [72, p. 21*3] on the
distribution along the real line of points in a periodic orbit of odd
period n > 1 , when there are no periodic orbits of odd period m with
1 < m < n , our results give a new proof of Sharkovskii's theorem.
REMARK 3.
Table 1 (see p. 96) l i s t s the f i r s t 31 values of
for 1 2 n 5 5 • I t seems t h a t , for a l l positive integers
have
$n(2nH-l)/(2m-l) = 2m~n
for
n
$ (m)/m
and m , we
n S m £ 3«+l ,
and
$ (2m+l)/(2m+l) > 2m~* for
REMARK 4.
For a l l positive integers
V(m) = $(m, ty) , where
m > 3n + 1 .
k
and m , l e t
* is defined as in Section 2.
i|j(fc) = 2
and
I t is obvious that
y(m)/m is the number of distinct periodic orbits of minimal period m
for, say, the mapping
£?(x) = kx(l-x)
from
and n with
[0, l ]
onto i t s e l f .
for a l l positive integers
k
1 2 k 5 2n ,
c,
= 1 , we obtain that
lw
$ (2fc+2)/(2fc+2) = ¥(fc+l)/(Zc+l)
rc
c ,
Since,
k+1
=2
- 1
for a l l
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96
Bau-Sen Du
TABLE 1
m
$Am)/m
• 2 (m)/»
$ (m)/m
¥<»>/«
1
1
1
1
1
1
2
1
1
1
1
1
1
3
1
0
0
0
0
2
1*
1
1
1
1
1
3
5
2
1
0
0
0
6
6
2
2
2
2
2
9
7
1*
2
1
0
0
18
8
5
3
3
3
3
30
9
8
1*
2
1
0
56
10
11
6
6
6
6
99
11
18
8
1*
2
1
186
12
25
11
9
9
9
335
13
1*0
16
8
l*
2
630
1U
58
23
18
18
18
1161
15
90
32
16
8
1*
2182
16
135
1*6
32
30
30
1*080
17
210
66
32
16
8
7710
18
316
56
56
1U56O
19
91*
136
61
1*92
61*
32
16
27591*
20
750
195
115
101
99
52377
21
1161*
282
128
6k
32
99858
22
1791
1*08
221*
191
186
190557
23
2786
592
258
128
61*
361*722
21*
1*305
856
1*31
351
337
698870
25
6710
121*8
520
256
128
131*2176
26
10U20
181U
850
668
635
2580795
27
16261*
261*6
1050
512
256
1*971008
28
25350
3858
1673
1257
1177
9586395
29
39650
561*1*
2128
1026
512
18512790
30
61967
821*6
3328
21*02
2220
35790267
31
97108
12088
1*320
2056
1021*
69273666
2
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Peri od i c orb i ts
1 5 k 5 2n . It seems that
97
$ (2k+2)/(2k+2) > 4'(k+l)/(k+l)
n
for all
k > 2n . But note that
lim (log[$n(2k+2)/(2k+2)])/(2k+2)
= log X > h log 2 = k lim
M
k-x»
where
A
is the unique positive zero of the polynomial
3. Symbolic representation for continuous piecewise linear functions
In this section we describe a method. This method was first
introduced in [4], and then generalized in [5] to construct, for every
positive integer n , a continuous piecewise linear function from [0, l]
into itself which has a periodic orbit of minimal period 3 , but with the
property that almost all (in the sense of Lebesgue) points of [0, l] are
eventually periodic of minimal period n with the periodic orbit the same
as the orbit of a fixed known period n point. The same method was also
used in [6] to give a new proof of a result of Block et at [/] on the
topological entropy of interval maps. In this paper we will use this
method to prove our main results.
Throughout this section, let g be a continuous piecewise linear
function from the interval [a, d] into itself. We call the set
{ [x., y .) | i = 1, 2, . .. , k} a set of nodes for (the graph of) y = g(ar)
if the following three conditions hold:
(1) k > 2 ,
(2) x± = a , x^= d , x± < x 2 < ... < a^ , and
(3) g is linear on
y. = g(x.)
[x., x. ]
for all 1 < i « k-1 and
for all 1 < i 5 k .
For any such set, we will use its ^-coordinates
represent the graph of y = g(x)
y , y , ..., y, to
and call i/,*^ ••• Dv (in
tnat
order) a
(symbolic) representation for (the graph of) y = g{x) . For
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98
Bau-Sen Du
1 si
< 3 5 k , we will call
y .y. . . . . y . the representation for
I' ^ ' J .
y = g(x)
Q
on [x., x.] obtained by restricting
*/..£/„ • •. J/, to
1 3
[x.,
x.J
a node.
.
i. c.
For convenience, we w i l l also c a l l every
If
y.
in
y,2/ o •
for some i (that i s , g i s constant on
simply write y± ... i / ^ + 2 • • • hk instead of
y. = j / .
we w i l 1
)>
K
£/•• ••• y-y • .->y • p ••• J/r. •
Therefore, every two consecutive nodes in a
(symbolic) r e p r e s e n t a t i o n are d i s t i n c t .
Now assume t h a t
y = g(x)
{ ( # . , J/.) | i = 1 , 2, . . . , k\
and a a~ ...
a
i s a representation for y = g(x)
{a , a , . . . , a } c \y , y
1 2 i 5 r-1 .
If
i s a s e t of nodes for
...,
y \
and a. £ a.
with
for a l l
{# , y , . . . , !/,} c {x , x , . . . , x , } , then there i s an
J-
c.
K
ci
X
K
2
easy way to obtain a representation for y = g (x) from the one
a a_ . . . a
for z/ = g(x) . The procedure is as follows. F i r s t , for any
two distinct real numbers u and v , l e t [u : v] denote the closed
interval with endpoints u and V . Then l e t b. b.
... b. . be the
representation for
restricting
a
y = ^(x)
a
-i o •••
a
to indicate this fact:
a. < a .
, or a .0..
to
on
Qz. : a.
[a. : a.
1 .
a .a.
-*• b.
b.
•*• b .
...
b. ob .
The above representation on
[a. : "•.•,]
{a l 5 a 2> . . . , a^} c {x 1 , x g , . . . , xfe} .
z.
. = b.
.
for a l l
J
l < i < t .
.
If
We use the following notation
... b.
(under
(under
g ) if
g ) if
<z . > a..
exists since
Finally, if
a. > a.
for all 1 5 ,7 5 t. . It is easy to see that
1 5 •£ 2 r-1 .
which is obtained by
n
z.
a^ < a^ +1 , l e t
, let
z.
= z.
for all
So, i f we define
then i t i s obvious t h a t
2
Z i s a representation for y = g (x) .
I t is
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P e r i o d i c orb i +s
99
also obvious that the above procedure can be applied to the representation
Z for
y = g (x)
to obtain one for
4.
In t h i s s e c t i o n we l e t
that is
fix)
= -2.x + 5
if
y = g ix) , and so on.
Proof of Theorem 1
fix)
denote t h e map as defined i n Theorem 1 ,
1 2 x 5 2 ,
and f(x)
= x - l
if
2 £ x £ 3 .
The proof o f p a r t (a) of Theorem 1 w i l l follow from two easy lemmas.
LEMMA 4 .
Under
f , we have
13 -»• 312 ,
31 -»• 213 ,
12 ^ 31 ,
21+13 .
In the following when we say the representation for y = j(x)
, we
mean the representation obtained, following the procedure as described in
Section 3, by applying Lemma h t o the representation
312 for y = fix)
successively u n t i l we get t o the one for y = j (x) .
For every p o s i t i v e integer
the number of
(vo v
-\ v
x-coordinates are £
respectively)
Up
=u
2 .
=0 .
LEMMA 5.
k
respectively) denote
(2 respectively)
whose corresponding
I t i s clear t h a t
u
2 .
We also l e t
1 2 ' s and 21's in the
x-coordinates are 5
= vo . = 1 and
Now from Lemma It, we have
1,1
c ,X
W
[uo ,
respectively) denote the number of
representation for y = jix)
(i
u. ,
1 3 ' s and 31's in the representation for y = j (x) whose
corresponding
v
k , let
= u
l,k
That is,
For every positive
+ v
+ U
u
s
l,k
( »,} ^
2,k >
then
the Lucas
w
1=
integer
1
'
k
W
2 =
' ^
w
k +2
= W
k+1
+ W
k •
k , the number of distinct
solutions of the equation j (x) = x equals
exclusion argument.
3
i = 1, 2 ,
sequence.
Since, for every positive integer
follows from Lemma 5.
and integers
w, , part (a) of Theorem 1
Part (b) follows from the standard inclusionAs for part (c) , we note that, for every positive
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100
Bau-Sen Du
integer
k ,
So, for k > 6 ,
where [(/c+3)/2] is the largest integer less than or equal to {k+3)/2 .
The proof of the other statement of part (a) is easy and omitted. This
completes the proof of Theorem 1.
5.
Proof of Theorem 2
In this section we fix any integer
map as defined in Theorem 2.
n > 1 and l e t f (x)
For convenience, we also l e t S
denote the
denote the
set of a l l these hn symbolic pairs: i{i+l), (i+l)i , 1 £ i 2 n-1 ;
n(n+2), (n+2)n, (n+l)(2n+l), (2n+l)(n+l), j ( j + l ) , (j+l)j , n+2 5 j 5 2n .
The following lemma is easy.
LEMMA
6.
Under
f
3
we have
n(n+2) •* (n+3)(n+2)n ,
(n+2)n -*• n(«+2)(n+3) j
) -»• (n+2)n(n-l)(n-2) . . . 321 ,
)
H-
123
...
and uv -+ f (u)f (v) for every
( (
uv in
Sn - {n(n+2), (n+2)n,
In the following when we say the representation for
y = j(x)
, we
mean t h e r e p r e s e n t a t i o n o b t a i n e d , following the procedure as described in
S e c t i o n 3 , by applying Lemma 6 t o the representation
l)
. . . (n+2)n(n-l)(n-2)
. . . 321
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P e r i od i c o r b i t s
for
10 1
y = f (x) successively u n t i l we get t o the one for y = 7 (x) .
For every p o s i t i v e integer
1 5 i , j 5 2n , l e t b, . .
k and a l l integers
i , 3 with
denote the number of uv's
the representation for y = j(x)
whose corresponding
and vu's
in
x-coordinates a r e
in [i, i + l ] , where uv = 3(3+1) i f 1 2 3 5 n - 1 or n+2 < j £ 2n ,
MU = n(n+2) i f 3 = n , and wu = (n+l)(2rc+l) i f j = n + 1 . I t i s
obvious t h a t i> ,
,
=1 , b . _ „ .
= l i f 2 < i « n ,
= 1 if
.
i,n
n+1 5 i S 2n , and £> . . = 0 elsewhere.
\,v,Q,n
Lemma 6, we see that the sequences
(bv • .
From
) are exactly the same as
those defined in Section 2.
Since
2n
e,
= V b, . .
fe." ^
k,z^,n
i t i s clear t h a t
0,
K,n
2n
+ b.
,
+ y fc, .
k,n+l,n,n
ki
i=£
i s the number of i n t e r s e c t i o n points of the graph
of
h ~ j(x)
2.
Part ("W follows from the standard inclusion-exclusion argument. As
with the diagonal
y =x .
for part (c), we note t h a t there e x i s t
This proves p a r t faj of Theorem
2n + 1 nonzero constants
a.'s
3
such that
for all positive integers
2n+l
b,
= > a .x.
1
k,l,n,n
H^ 3 3
3=
k , where {x. | 1 £ j £ 2n+l} is the set of
a l l zeros (including complex zeros) of the polynomial
Since e,
can also be expressed as
K"T£lYld
b
x
- 2x
- 1
n
k+2n-2,n+l,n,n T ""fc.l.n.n T > o
part fcj follows from property (i) of the sequences
(b, . . \ stated in
Section 2. This completes the proof of Theorem 2.
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102
Bau-Sen Du
References
[7]
L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young,-"Periodic
points and topological entropy of one dimensional maps", 18-31*
(Lecture Notes in Mathematics, 819.
Springer-Verlag, Berlin,
Heidelberg, New York, 1980).
[2]
R. Bowen and J. Franks, "The periodic points of maps of the disk and
the i n t e r v a l " , Topology 15 (1976), 337-31*2.
[3]
U. Burkart, "Interval mapping graphs and periodic points of
continuous functions", J. Cornbin. Theory Ser. B 32 (1982),
57-68.
[4]
Bau-Sen Du, "Almost a l l points are eventually periodic with minimal
period
[5]
3 " , Bull.
Inst.
Math. Aaad. Siniaa 12 (198M, l^- 1 *!!.
Bau-Sen Du, "Almost a l l points are eventually periodic with same
periodic orbit" (Preprint, Academia Sinica, Taiwan, Republic of
China, 1981*).
[6]
Bau-Sen Du, "The periodic points and topological entropy of interval
maps" (Preprint, Academia Sinica, Taiwan, Republic of China,
198U).
[7]
C.-W. Ho and C. Morris, "A graph theoretic proof of Sharkovsky's
theorem on the periodic points of continuous functions",
Pacific
J. Math. 96 (1981), 361-370.
[8]
L. Jonker, "Periodic points and kneading invariants", Proa. London
Math. Soc. (3) 39 (1979), 1+28-1*50.
[9]
Tien-Yien Li and James A. Yorke, "Period three implies chaos", Amer.
Math. Monthly 82 (1975), 985-992.
[10]
M. Misiurewicz, "Structure of mappings of an interval with zero
entropy", Inst.
[II]
Hautes Etudes Sci. Publ. Math. 53 (1981), 5-l6.
A.N. Sharkovskii, "Coexistence of cycles of a continuous map of a
line into i t s e l f " , Ukrain. Mat. 2. 16 (196U), 6l-71.
[12]
P. Stefan, "A theorem of Sharkovsky on the existence of periodic
orbits of continuous endomorphisms of the real l i n e " , Comm.
Math. Phys. 54 (1977), 237-2U8.
Downloaded from https://www.cambridge.org/core. IP address: 88.99.165.207, on 31 Jul 2017 at 13:42:53, subject to the Cambridge Core terms of use,
available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700002306
Peri od i c orb i ts
103
[13] P.D. Straff in, Jr., "Periodic points of continuous functions", Math.
Mag. 51 (1978), 99-105.
Institute of Mathematics,
Academia Sinica,
Nankang,
Ta i pe i,
Ta i wan 115,
Republic of China.
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available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700002306