On an extension of Sarkovskii's order

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

On an Extension

APPLICATIONS

138,

52-58

of Sarkovskii’s

(1989)

Order

L. A.V. CARVALHO* Instituto

de Ci&cias Matemciticas de Xio Carlos, Universidade de Silo Paula, Au. Dr. Carlos Bolelho, 1465, 13.560, Scio Carlos, S.P., Brazil Submitted Received

by Kenneth August

L. Cooke 1, 1987

In 1964, A. N. Sarkovskii proved a fundamental result concerning the continuous maps of the interval. It is as follows: let J denote an interval in R, the real line; J can be open, half open, closed, of finite or infinite length and, in particular, it can be R itself; consider a continuous map f: J-+ J and for a given x E J consider the sequence O+(x) = def(x, f(x), f’(x), ...} consisting of x and its successive iterates under the action of f; O+(x) is called “the orbit of x” and, viewed as set, it is either finite or infinite. It is finite if, and only if, there exists a natural number n > 1 such that f”(x) = x; in this case, the least such n is called “the period of x” or, equivalently, “the period of C+.(x).” When n = 1, x is just a fixed point of f; when n 2 1 we shall say that “x is a n-periodic point (off)” or, equivalently, that “Lof(x) is a n-periodic orbit”; in particular, x is n-periodic with n > 1 if, and only if f”(x) = x and f’(x) #x for 1 ,
1 Q 2 U 4 U 8 4 .. . 4 7.2’ U 5.2’ 4 3.2’ 4 . . . u . . . a 72wherej=1,2,3

1 a 52-l

a 32-l

supported by FAPESP-Funda$o de Amparo B Pesquisa do while the author was on leave at Brown University, Providence,

52 Copyright All rights

(*I

,....

*This work was partially Estado de Slo Paula-Brazil, RI (1983/1984). 0022-247X/89

Q . . . a 7 a 5 a 3,

$3.00

0 1989 by Academic Press, Inc. of reproduction in any form reserved.

EXTENSION

OF SARKOVSKII’S

53

ORDER

Let’s colloquially comment a little bit on this. First of all, (*) is composed of three different kinds of groups of natural numbers: the powers of 2, the odds times a same positive power of 2, and the odds. The group of powers of 2 comes first, ordered according to the regular increasing magnitude order; the group of odds comes last, ordered according to the regular decreasing order and, in between, one puts, each group at a time, the groups of odds times a power of 2 ordered in the following fashion: each group belonging to the same power of 2 is ordered according to the regular decreasing order of the odds and the groups themselves are ordered according to the regular descending order of the exponents of 2. This is, doubtless, an easthetically beautiful result. Besides, it has proved to be very useful in the ellucidation of various questions related to the complex behaviour of certain discrete and continuous dynamical systems [2-I]. Let’s now make a sharper scrutiny of the periodicity phenomenon described above. Let x be a n-periodic point of f: We shall put x, = min Co/ix) and x,=f(x,), x3 =f(xr), .... x,= (f(x, ,)). With this convention, there are, altogether, (n - l)! possibilities for such an orbit @+,) = (XI, x2, ...?x,), depending solely on the relative position of the points x2, x3, .... x,,. In order to fix the ideas, let n = 4 and consider the six different types of 4-periodic orbits that f might have as shown in Fig. 1. Suppose then that one knows that f has, say, an orbit of period 7. It is apparent that it is impossible to make an a priori decision about the types of, say, the 4-periodic orbits that must exist, solely on the basis of Sarkovskii’s theorem. This is so because various, or even all, types of orbits of period n may coexist for the same map f, depending exclusively in its shape. + 1 : : I :a

__----

: _---o----t-

,

I

I,

x *

_ vx3

2

54

L. A. V. CARVALHO

FIGURE

2

In order to better explain this idea, observe that there is only one type of l-periodic orbit, only one type of 2-periodic orbit, and only two types of 3-periodic orbits. These latter are shown in Fig. 2. Of course, we may arrange things so that all six 4-periodic and both 3-periodic orbits coexist for the same f: In this procedure, it is advisable that one departs from previously given orbits and then find the candidate f: It goes in a direction somewhat opposite to the spirit of Sarkovskii’s theorem but is very useful to furnish examples and counter examples. So, suppose we are given a unimodal map f with an orbit of period 3. A unimodal map is a continuous map of the interval such that there exists an interior point 5 at the interval with f strictly increasing for x < 5 and strictly decreasing for x > 5. The given 3-periodic orbit can only be, then, the orbit x,
f(4=(x-'i;j;,

7

;::z; \ \ .

Then f is unimodal and @$(l)= (1,2,3) IS . a 3-periodic orbit for f of the type of Fig. 2A. Now, for any x E [ 1, 31, we have that either f(x) < x or f *(xl Gf (x). H ence there is no 4-periodic orbit of the type x < f(x) < f*(x)
EXTENSION

OF SARKOVSKII’S

Fig. IA belong to a singular family element of order n is of the form

ORDER

of periodic

qi(x1, = 1x1,x* )..., x,:x,
55

orbits,

whose generic

...
The easiest to check feature of this kind of orbits is the fact that they are the only orbits with just one point below the diagonal (x, x). This feature allows us to, unambiguously use just the symbol “n,” to denote the generic element of the family. Thus the orbit of Fig. 2A is a 3, orbit, the orbit of Fig. 1A is a 4, orbit, while, as we said, the 2-periodic orbit and l-periodic orbit are typically unique, we shall not need to use this special symbol for them. Sometimes, we shall also refer to a n, orbit as a n-step orbit. Observe, also, that the reasoning we did before to show that if the “3” in (*) is “3,” then the “4” is not “4, ,” is readly extendable to show that, indeed, if the “3” in (*) is “3 1,” then any n > 3 in (4 is not “n, .” And, because of this, we can reserve the locution “n-orbit” to unambiguously denote a n-periodic orbit that is not a n,-orbit. Consider now the set N’ = { 1, 2, 4, 5, 6, .... 3,) 4,) 5,) ... ), and let relation interval, it. Then.

p, q, Y be variables assuming values in N’. Define Sarkovskii’s on N’:p Q q if, and only if, for each continuous map of the the existence of a q-orbit for it implies the existence of a p-orbit for we have:

THEOREM. The above defined relation is a total order relation on N’ and, under it, the ordering of N’ is

la24448a... 4 . . . q 7.2’1

Q 7.2’ 0 5.2’
a “’ 0 7 4 5 a 3, u 4, c 5, .“,

(**I

where j = 1, 2, 3, ...

Observe that the theorem is valid for the chain 1 U 2 <1 4 a . G . . C 7 Q 5 Q 3, due to the above remarks on Sarkovskii’s theorem. We shall, therefore, prove it just for the chain 3, a 4, U 5, 4 . . . The proof will be given first to unimodal maps and then it will be extended to the general case. We need the following definition: a continuous map of the interval is said to have property Sk, k>l, at a point x if x
56

L.A.V.CARVALHO LEMMA.

If f

is unimodal and has a n,-orbit {x,x2 ... xn}, n > 3, then

f E Snpi(xi), j= 1, 2, . ... n - 1. ProoJ: Indeed, the monotonicity property of f implies that [x,, xi+ ,] = [xi+ ,, xj+J, j= 1,2, .... n - 2, where [a, b] denotes the closed interval a < x < 6. Hence, if y E [x,, xj+ i ] we have that

f

Y

< ... -=f”-‘-‘(y),

which proves the lemma. In particular, we have that f ES,_2(x,). So, suppose a unimodal f has a p-step orbit, x1 < x2 < ... 3. Refer to Fig. 3, Let y = max f -‘(x2). Then, YE (x,_ ,, x,) and f(y) =x2. So, we have at once

Hence, the continuity of f implies, via Bolzanos’s intermediate value theorem, the existence of a z E (y, xP) such that fp- '(z) = z. Let zi = f(z), Then zi E (xi, x2). Since f ESp-I(x,) it follows immediately from the lemma that

ZI
...
which readily shows that O+(z,) is a (p - l), orbit. This finishes the proof for the case of a unimodal map. In order to prove the theorem for the general case, suppose that a n

FIGURE 3

EXTENSION

OF SARKOVSKII’S

ORDER

57

continuous map J of the interval has a n-step orbit x1 < x2 < . .. < x,,, n > 3. We define the companion map f off relative to this orbit as f

if x6x, if x,
x2

;;.y 1 x3

;;4 f(x)=,

;” X,-I f(x) X, X,

andf(x)
if x “-2dx
f(x) \ XI

and f(x) Q x, ~ l and x,-,
Note that X is well defined due to the continuity off: It is also clear that f is a continuous map of the same interval as f and that, by construction, { XI, x2, .... x,> is a n-step orbit off as well. Refer to Fig. 4 in order to have a pictorial idea oft It is easy to see that

P~cxi-,~xil)= As a consequence, y=max{,j‘~‘(x2)n(x,~

i=2,

cxi?x,+Ll,

3, .... n- 1.

it follows that YES,,-,(x,). So, if we take ,, x,,)}, the above proof given for unimodal maps

FIGURE

4

58

L. A. V. CARVALHO

can be literally transposed to the present situation, in order to obtain a (n- 1)i orbit of f, z1
Consider the unimodal

map

fA: [ - 1, 1] + [ - 1, 1 ] given by

f(x) = 1 - Ax2 for values of the parameter K in the interval (0,2].

It can be shown [2] that the orbit 3, appears at A= 1, 75 the orbit 4, at A = 1.941 .... the orbit 5, at 1.985 .... the orbit 6, at 1.996 .... .... and that, prior to A= 1,75, the whole Sarkovskii sequence (and much more) is present. Some remarks are now due:

(1) It seems clear from the above results that the complete form of Sarkovskii’s order, in case it exists, should be searched for in the set consisting of all permutations of the natural numbers { 1,2, .... n}, n = 1, 2, 3, .... which leave 1 fixed. (2) If we denote by “nl” the n-periodic orbit x1 > x2 > . . . > x,, n 2 3, which is the generic element of the family determined by the 3-periodic orbit shown in Fig. 2B, then, obviously, the order that one obtains from (w) by substituting “n,” by “nl” is valid. (3) There is an alternative proof for Theorem 1, as Professor Z. Nitecki had personally shown to author in 1984. This proof uses the symbolic techniques developed by Li, Yorke, Block, Misiurewicz, and others [3]. (4) In a forthcoming paper, the author will analytically show that the n,-orbits are the last stable periodic orbits of the logistic map f).(x) = 2x( 1 -x), for the parameter 1 ranging in the interval (0,4].

REFERENCES A. N. SARKOVSKII, Coexistence of cycles of a continuous map of the line into itself (in Russian), Ukruin. Mat. Zh. 16 (1964), 61-71. 2. P. COLLET AND J.-P. ECKMANN, “Iterated Maps on the Interval as Dynamical Systems” (A. Jaffe and D. Ruelle-Birkhluser, Eds.), PPh Series, Boston, 1980. 3. L. P. KADANOFF, “Roads to Chaos,” Physics Todq, December, 1983. 4. Z. NITEKI, Topological dynamics on the interval, in “Ergodic Theory and Dynamical Systems Proceedings of the Special Year, 1981-PM Series,” Birkhluser, Boston, 1981. 1.