ON THE NUMBER OF PERIODIC ORBITS OF CONTINUOUS MAPPINGS OF THE INTERVAL

Let f be a continuous map of a closed interval into itself, and let P(f) denote the set of positive integers k such that f ha$ a periodic point of period k . Consider the following ordering of positive integers : 3,5,7, . . . . 2 .3,2 .5,2 .7, . . . . 4 .3,4 .5,4.7, . . .,8,4,2,1 . Sarkovskii's theorem states that if n e P(f) and m is to the right of n in the aboye ordering then m E P(f) . We may ask the following question : if n E P(f) and m is to the right of n in the aboye ordering what can be said about the number of periodic orbits of f of period m ? . We give the answer to this question if n is either odd or a power of 2 .


.Introduction
This paper is concam ed with the periodic orbits of continuous mappings of the interval into itself .Let I denote a closed interval on the real line and let C o (I,I) denote the space of continuous maps of I into itself .For f E C o (I,I), let P(f) denote the set of positiva integers k such that f has (at least) a periodic pcint of period k (sea section 2 for definition) .One may ask the following question : If k E P(f), what other integers must be elements of P(f) ? .
This question is answered by a theorem of Sarkovskii .Consider the following ordering of the set of positiva integers N : Thus, in this ordering the smallest element of N is 3 and the greatest is 1 .Sarkovskii's thecrem states that if n C P(f) and m is.t o the right of n in the above ordering (Sarkovskii ordering) then there is , at least one periodic orbit of period m (seel21 or i3J) .Furthermore, if m is to the left of n in the Sarkovskii ordering, then there is a map f E C 0 (I,I) with n E P(f) and m 1 P(f) .
For f C C 0 (I,I), let N(f,m) denote the number of periodic orbits of f of period m .In this papar, we ask the following question : If n E P(f) and m is to the right of n in the Sarkovskii ordering, what can be said about N(f,m) ? .Our main result is the following .(i) There is en integer N (m) ( easily computable, sea section 3) ( ü~There is a map g E C (I,I) such th at P(g) = P(f) and N(9,m) = Nn(m) .
Note, for example, that if f E CQ (I,I) and 3 E P(f), then f has at least N 3 (m) periodic orbits of period m .We have compute N 3 (m) and N5(m) for m = 1,2, . ..,50 in Tablas I and II, respectively (for details sea section 3) .We remark that Sarkovskii's theorem only says N (M) -,> 1 .n Proposition B .Let f e. Co (I,I) and let n denote the minimum of P(f) in the Sarkovskii ordering .Suppose n is a power of 2 and m is to the right of n in the Sarkovskii ordering .Then the integer N n (m) which satisfies conditions (i) and (ii) of Theorem A is the unity .
In proving Theorem A, we use a result of Stefan (sea section 2) .
This result describes how a mapping f e C o (I,I) must act on a periodic orbit ip 1 1 . ..,pn 1 of odd period n > 1, where n is the minimum of P(f) in the Sarkovskii ordering .
We note the algorithm described in order to compute the integer N n (m) defined in Theorem A (sea section 3) can be used for all n e P(f) not necessarily odd .But we need to know how f must act on a periodic orbit of f of period n .That is, if )p1, . ..,pn1 is a periodic orbit of f of period n, who is f(p i ) for each i = 1, . . . .n ? .
We are grateful to Ramon Reventós who have helped us in the preparation of this note .

.Preliminary defin itions and results
Let f E C 0 (I,I) .For any positive integer n, we define fn inductively by f 1 = f and fn = f ^f n-1 .We let f denote the identity map of I .
Let p E I .We say p is a fixed point of f if f(p) = p .If p n is a fixed point of f , for soma nE N, we say p is a periodic point of f .In this case the smallest element of j n EN : fn(p) = p } is called the period of p .n , p) define the orbit of p to be , f (p~ : n = 0,1,2, . . .j .If p is a periodic point of f of period n, we say the orbit of p is a periodic orbit of period n .In this case the orbit of p contains exactly n points each of which is a periodic point of period .n .
Theorem A .Let f cy C o (I,I) and let n denote the mínimum of P(f) in the Sarkovskii ordering .Suppose n is odd, n 7 1 and m is to the right of n in the Sarkovskii ordering .Then the following hold .

Theorem 1 .
fig .1 Fig . 2 points of g are the points of the grephic of a13(m)+b23(m)+b13(M) = ( 1+ 2J5 ) m + ( 1-~5 ) m is one of the following for each i = 1, . ..,k(m)-1 .have that an d Let a rs (m) (respectively b rs (m), cr s (m), d rs (m» be the number of intervals[g i , q i+11 L Ip 1 ' p21 (respectively [p2' p31 ' U p 3' p 4]' [p4' PSI) such that g m ( [gi'qi+1] ) = [pr'p1 .From the definition of g we for m = Because the fixed points of gm are the points of the graphic of g m which are on the diagonal of the square (p 1 ,p5 ] x Ep1'p53