Periodic Points of Period Three
In the previous section , we discovered values 𝜇 for 𝑄𝜇 has fixed points and
periodic point of period 2𝑘 for all positive integer 𝑘 less than any given positive
integer 𝑛 , but no other periodic points.
In the present section , we will describe what the presence of a periodic point of
period 3 , or any periodic point of period 𝑛 ,implies about the existence of
periodic points .the answer will derive from two famous theorem, those of LiYorke (1975) and Sarkovskii (1964).
Our first goal will be to introduce a wonderful theorem due to James Yorke and
his student Tien Li. It tell us that if 𝑄𝜇 has a periodic point of period 3 , then 𝑄𝜇
has a periodic point of period 𝑛 ,for every 𝑛 ≥ 1.
Theorem (Li –York’s Theorem ):
Suppose that f is continuous on the closed interval 𝐽, with 𝑓(𝐽) ⊆ 𝐽 . if f has a
periodic point of period 3 . then f has periodic points of all other periods.
The Li –York’s theorem says that if 𝑓 has a periodic point of period 3 , then it
has points of all periods. But suppose that we can only show 𝑓 has, say,a periodic
point of period 5. Then must f have points of all period? A remarkable theorem
by the Russian mathematician A.N.Sarkovskii a complete answer. In order to
present Sarkovskii’s result, we need to define the Sarkovskii ordering of the
positive integers:
3∆5∆7∆ …
𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠
6 ∆ 10 ∆ 14 ∆ …
2(𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠)
12 ∆ 20 ∆ 28 ∆ …
4(𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠)
…
… ∆ 8 ∆ 4 ∆ 2 ∆ 1.
𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 2
Here m ∆ n signifies that 𝑚 appers before 𝑛 in the Sarkovskii ordering. The
17 ∆ 14 (because 14 = 2.7) and 40 ∆ 64 (because 64 = 26 )
Since every positive integer can be written as 2𝑘 (odd integer) for a suitable noninteger 𝑘 and a suitable odd integer, the Sarkovskii ordering is an ordering of the
collection of all positive integers.
Now, we are ready for theorem :
Theorem ( Sarkovskii’s Theorem)
Let 𝑓 be a continuous function defined on the interval 𝐽 and suppose that
𝑓(𝐽) ⊆ 𝐽 . If 𝑓 has a point with period m, then 𝑓 has a point with period 𝑛 for
all n such that 𝑚 ∆ 𝑛.
The original proof of this theorem was long and technical.
By letting 𝑚 = 3 in the Sarkovskii’s theorem, we see that the Li-Yorke theorem
.