Martin Heyde Lawaetz, Aktuel Matematik, 2008
The Sharkovski Theorem
Introduction
The Sharkovski Theorem is also known as The Sharkovsky Theorem. It depends upon which
articles you read. The mathematician who is known for the theorem is either called Alexandr
Nicolaevich Sharkovski or Oleksandr Mikolaiovich Sharkovsky. Some articles might even spell his
name otherwise. In the following I’ll call the theorem The Sharkovski Theorem and call him
Alexandr Nicolaevich Sharkovski.
Today the theorem is one of the classical results in the theory of dynamical systems and is known to
many mathematicians specializing in other areas [Ciesielski et al, 2008].
Alexandr Nicolaevich Sharkovski, Biography
Alexandr Nicolaevich Sharkovski was born on December 7, 1936, in Kiev, Ukraine. He graduated
from the local University of Kiev in 1958, where he also taught from 1967. He is today a prominent
mathematician and a member of the National Academy of Science of Ukraine.
Sharkovski’s main area of interest is the theory of dynamical systems, the theory of stability and the
theory of oscillations. He also works with the theory of functional and functional differential
equations, and the study of difference equations and their application.
History
In 1964 a paper written by Alexandr Nicolaevich Sharkovskii appeared in the Ukrainian
Mathematical Journal. The title of the paper was Coexistence of cycles of a continuous mapping of a
line into itself and it was 11 pages long. He already submitted the paper in 1962, when he was only
25. The purpose of the paper was to prove the following theorem: If k ◄ l, and if a continuous
mapping of the reals into the reals has a point with a fundamental period k, then it also has a point
with fundamental period l; ”◄” denote an order defined by Sharkovski, the Sharkovski’s ordering
[Ciesielski et al, 2008]. This theorem was later to be known as The Sharkovski Theorem.
In 1962 he had already published some results that can be regarded as an introduction to the
theorem.
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Martin Heyde Lawaetz, Aktuel Matematik, 2008
Sharkovski’s work did not become known outside Eastern Europe until the second half of the
1970s. The reasons why the paper and his work went unnoticed could be several. First of all it was
written in Russian and published in a Soviet mathematical journal. Another reason could be that the
topic wasn’t fashionable at the time [Ciesielski et al, 2008].
In 1975 American Mathematical Monthly published a famous paper with the title Period three
implies chaos by Tien-Yien Li and James A. Yorke. They proved a special part of Sharkovski’s
result besides introducing the idea of chaos. Because of the paper they attended a conference in East
Berlin where they met Sharkovski. Although they had no language in common the meeting led to
global recognition of Sharkovski’s work.
Another paper that brought Sharkovski’s work to attention of many mathematicians where the
paper: A theorem of Sharkovski on the existence of periodic orbits of continuous endomorphisms of
the real line published by the Slovak mathematician Peter Stefan [Ciesielski et al, 2008].
As many papers and conferences devoted to iterations, chaos and related phenomena appeared in
the late 1970s many people started to work on Sharkovski’s theorem. One of the initial goals was to
simplify his proof; many new and shorter proofs appeared. All of them relied largely on the
intermediate theorem [Ciesielski et al, 2008]. Around 1980, at least 3 proofs of the theorem were
published, all fairly similar. These constitute the modern ”standard proof”, due to Block,
Guckenheimer, Misiurewicz and Young, Burkart, Ho and Morris.
Today The Sharkovski Theorem is a standard mathematical term and Sharkovski is one of few
living mathematicians whose names are similarly attached to one or another of their results.
The theorem
Sharkovski’s theorem involves an ordering of the positive natural numbers, N, which is known as
the Sharkovski ordering.
Before presenting Sharkovski’s ordering we ask ourselves what an ordering is? We are used to the
ordering 1<2<3<4<........., but this is not the only way we can order the natural numbers. An
ordering is a list of the numbers, which has to satisfy a few conditions. For a pair of numbers (a, b)
in the list, only one of the following is true; ’a is before b’, ’a is equal to b’ or ’b is before a’. If ’a is
before b’ and ’b is before c’ then ‘a is before c’.
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Martin Heyde Lawaetz, Aktuel Matematik, 2008
The following ordering of the positive integers is known as the Sharkovski’s ordering:
3, 5, 7, 9, ..., 2∙3, 2∙5, 2∙7, ..., 22∙3, 22∙5, 22∙7, ..., 24, 23, 22, 21, 20 = 1.
(*)
We write l ► m or m ◄ l if l is to the right of m.
The list starts with the odd numbers in increasing order beginning with 3. The sequence is then
repeated with each odd integer multiplied by 2. Then the initial sequence is multiplied by 22=4, then
23=8, then 24=16 and so on. At the end we put the powers of two in decreasing order.
When you look at the list you might wonder what this ordering is for? But this ordering is an
indispensable component of The Sharkovski Theorem. Sharkovski showed that this ordering
describes which numbers can be least periods for a continuous map/function.
Suppose f: R → R is a continuous function. We say that the number x is a periodic point of period
m if fm(x) = x, ”fm” denotes the iteration of our mapping/function; f2(x) = f(f(x)), f4(x) = f(f(f2(x))).
In other words we can say a point is periodic if it returns to its initial position after a definite
number of iterations. Specifically, if a periodic point x after m iterations returns to its initial
position, then fm+1(x) = f(x) [Ciesielski et al, 2008]. Furthermore we say that the number x have a
least period m if fk(x) ≠ x for all 0<k<m. This is also called the fundamental period of x.
It should be obvious that if m is a period of a number or a point then so is any multiple of m.
If a number has a period 1 then this number or point is called a fixed point.
Besides periodic points we also have a class of non-periodic points. These points never return to
their initial position: fn(x) ≠ x, for all n  N.
We will give some examples:

Consider the function g(x) = x. All points for the function g are fixed points.

Consider the function h(x) = 2x. Does this function have any fixed points or any points with
period 2, 3, and 4?? ...Here 0 is a fixed point, since h (0) = 2∙0 = 0 and there are no other
periodic points, all other points are non-periodic.

Consider the function j(x) = -x. This function has 0 as a fixed point and all other points are
periodic with fundamental period 2: for x = 5: j(5) = -5.....j2(5) = j(j(5)) = j(-5) = 5.
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Martin Heyde Lawaetz, Aktuel Matematik, 2008
In all of the functions above it is easy to find periodic points and their fundamental period. If we
look at the function f(x) = k∙x (1 – x) it gets more interesting. If k = 3½, then we have points with
least periods 1, 2 and 4 [Ciesielski et al, 2008]. Can we find a function with the fundamental period
3?
Lets find a continuous function defined for all real numbers with f(1) = 3, f(2) = 1 and f(3) = 2. We
draw the graph of such a function, figure A:
When we look at the graph it should be clear that the value 1, 2 and 3 has a least period 3.
In the above we’ve been interested in finding periodic points of a function f and the possible
periods. The Sharkovski Theorem is here very helpful.
The Sharkovski Theorem states that if a function f has a periodic point with fundamental period m
and m ◄ n in the ordering given above (*), then f has also a periodic point with fundamental period
n. This means that if f has a point of period 2 then we can state that f also has a point of period
1(fixed point), but we cannot state anything about points with period 4. Furthermore, if there is a
periodic point of period three, then there are periodic points of all other periods.
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Martin Heyde Lawaetz, Aktuel Matematik, 2008
The theorem also give us that if k ◄ n, then we can find a function that has a point with
fundamental period n but none with k.
Earlier we made an assumption of continuity, which is very important. Look for example at the
discontinuous function f: x → (1 - x)-1. Every value has period 3, which would be a
counterexample.
Furthermore it shall be pointed that Sharkovski’s result is one-dimensional. Consider a rotation of a
plane through 120°. Iteration of this rotation has the center of rotation as a fixed point and all other
points in the plane are periodic with fundamental period 3.
I will not prove the theorem in this report, but only give a little information about Sharkovski’s
proof and later proofs. Sharkovski’s proof contains no mathematical ‘fireworks’ and do not use
advanced results but relies on the repeated use of the intermediate value property. Before publishing
the actual theorem he showed that if a continuous mapping has a point with fundamental period 2n,
then it also has points with fundamental period 2n-1, 2n-2, ..., 21. He also proved that if k is not a
power of 2 and you can find a periodic point with a least period k then you can also find a periodic
point with a least period 2n, for all n [Ciesielski et al, 2008].
In later modern proofs of The Sharkovski Theorem you usually find three basic steps. If we denote
the property: ‘if f has a point with fundamental period k, then it has a point with fundamental period
n’ by ‘k ||- n’, then the 3 steps are as follows. First it’s shown that n ||- 2 for n > 2, next it’s shown
that k ||- n for all odd k and 3  k < n. Finally k ||- n is shown for all odd k and even n. When you
then apply these steps to f and f’s iterates it leads to the theorem [Ciesielski et al, 2008].
Li and Yorke (1975) proved that the presence of a periodic point of period 3, in a continuous
mapping of a closed interval onto itself, implies the presence of periodic points of all other periods
[Burns et al, 2007]. It should be clear that this is just a part of Sharkovski’s result.
They also proved that the existence of a point with a least period 3 implies the existence, in the
interval, of a ”large” set of points that never return to their initial under iteration, and behave in a
very erratic and chaotic manner [Ciesielski et al, 2008].
The theorem is as stated in the beginning a classical result in the theory of dynamical systems
because the ingredients of a dynamical system is a set and the mapping of this set into itself. If a
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Martin Heyde Lawaetz, Aktuel Matematik, 2008
dynamical system describes a certain phenomenon or process, it is important to know that a specific
concrete situation will repeat itself [Ciesielski et al, 2008].
Generalizations
When the paper by Sharkovski was published in 1964 it went unnoticed but in the last 30 years his
results has been attracting attention of many mathematicians. A number of people have suggested
new versions of the proof as mentioned. Other mathematicians have investigated the possibility of
extending the results to more complex structures, to broader classes of maps (discontinuous, multivalued and so on), to different types of phase spaces (one-dimensional: circle, stars or even
multidimensional and infinite-dimensional). All these new studies have led a rise of a new section
in the Mathematics Subject Classification of AMS [www.scholarpedia.org].
References
Burns K & Hasselblatt B; Sharkovsky’s Theorem, March 2, 2007, s. 1-14.
Ciesielski K. & Pogoda Z.; On Ordering the Natural Numbers or The Sharkovski Theorem, The
Evolution of....edited by A. Shenitzer & J. Stillwell; American Mathematical Monthly, February
2008, 159-165.
Nørgård-Sørensen Sune & Falsled Mikkel; Kaotiske systemer, Almen Matematisk Dannelse,
Matematisk afdeling KU, Forår 2002, s. 5-15.
Websites:
[http://www-history.mcs.st-andrews.ac.uk/Biographies/Sharkovsky.html]
[http://mathworld.wolfram.com/SharkovskysTheorem.html]
[http://en.wikipedia.org/wiki/Sharkovskii%27s_theorem]
[http://www.scholarpedia.org/article/Sharkovsky_ordering]
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