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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
has a
periodic point In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a ...
of period 3, then it must have periodic points of every other period.


Statement

For some interval I\subset \mathbb, suppose that f : I \to I is a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
. The number x is called a ''periodic point of period m'' if f^(x)=x, where f^ denotes the
iterated function In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...
obtained by composition of m copies of f. The number x is said to have ''least period m'' if, in addition, f^(x)\ne x for all 0. Sharkovskii's theorem concerns the possible least periods of periodic points of f. Consider the following ordering of the positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s: \begin 3 & 5 & 7 & 9 & 11 & \ldots & (2n+1)\cdot2^ & \ldots\\ 3\cdot2 & 5\cdot2 & 7\cdot2 & 9\cdot2 & 11\cdot2 & \ldots & (2n+1)\cdot2^ & \ldots\\ 3\cdot2^ & 5\cdot2^ & 7\cdot2^ & 9\cdot2^ & 11\cdot2^ & \ldots & (2n+1)\cdot2^ & \ldots\\ 3\cdot2^ & 5\cdot2^ & 7\cdot2^ & 9\cdot2^ & 11\cdot2^ & \ldots & (2n+1)\cdot2^ & \ldots\\ & \vdots\\ \ldots & 2^ & \ldots & 2^ & 2^ & 2^ & 2 & 1\end It consists of: * the odd numbers = (2n+1)\cdot2^0 in ''increasing'' order, * 2 times the odd numbers = (2n+1)\cdot2^1 in ''increasing'' order, * 4 times the odd numbers = (2n+1)\cdot2^2 in ''increasing'' order, * 8 times the odd numbers = (2n+1)\cdot2^3, * etc. = (2n+1)\cdot2^m * finally, the powers of two = 2^n in ''decreasing'' order. This ordering is a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
: every positive integer appears exactly once somewhere on this list. However, it is not a
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-orde ...
. In a well-order, every subset would have an earliest element, but in this order there is no earliest power of two. Sharkovskii's theorem states that if f has a periodic point of least period m, and m precedes n in the above ordering, then f has also a periodic point of least period n. One consequence is that if f has only finitely many periodic points, then they must all have periods that are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods. Sharkovskii's theorem does not state that there are ''stable'' cycles of those periods, just that there are cycles of those periods. For systems such as the
logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
, the
bifurcation diagram In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the syst ...
shows a range of parameter values for which apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer-generated picture. The assumption of continuity is important. Without this assumption, the discontinuous piecewise linear function f:[0,3) \to [0,3) defined as: f: x\mapsto \beginx+1 &\mathrm 0\le x<2 \\ x-2 &\mathrm 2\le x< 3\end for which every value has period 3, would be a counterexample. Similarly essential is the assumption of f being defined on an interval. Otherwise f : x \mapsto (1 - x)^, which is defined on real numbers except the one: \mathbb R\setminus\, and for which every non-zero value has period 3, would be a counterexample.


Generalizations and related results

Sharkovskii also proved the converse theorem: every upper set of the above order is the set of periods for some continuous function from an interval to itself. In fact all such sets of periods are achieved by the family of functions T_h:[0,1]\to ,1/math>, x\mapsto\min(h,1-2, x-1/2, ) for h\in ,1/math>, except for the empty set of periods which is achieved by T:\mathbb R\to\mathbb R, x\mapsto x+1. On the other hand, with additional information on the combinatorial structure of the interval map acting on the points in a periodic orbit, a period-n point may force period-3 (and hence all periods). Namely, if the orbit type (the cyclic permutation generated by the map acting on the points in the periodic orbit) has a so-called stretching pair, then this implies the existence of a periodic point of period-3. It can be shown (in an asymptotic sense) that almost all cyclic permutations admit at least one stretching pair, and hence almost all orbit types imply period-3.
Tien-Yien Li Tien-Yien Li (李天岩) (June 28, 1945 – June 25, 2020) was a University Distinguished Professor of Mathematics and University Distinguished Professor Emeritus at Michigan State University. There, he spent 42 years and supervised 26 Ph. ...
and James A. Yorke showed in 1975 that not only does the existence of a period-3 cycle imply the existence of cycles of all periods, but in addition it implies the existence of an uncountable infinitude of points that never map to any cycle ( chaotic points)—a property known as
period three implies chaos James A. Yorke (born August 3, 1941) is a Distinguished University Research Professor of Mathematics and Physics and former chair of the Mathematics Department at the University of Maryland, College Park. Born in Plainfield, New Jersey, United ...
. Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces. It is easy to find a
circle map In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamica ...
with periodic points of period 3 only: take a rotation by 120 degrees, for example. But some generalizations are possible, typically involving the mapping class group of the space minus a periodic orbit. For example,
Peter Kloeden Peter may refer to: People * List of people named Peter, a list of people and fictional characters with the given name * Peter (given name) ** Saint Peter (died 60s), apostle of Jesus, leader of the early Christian Church * Peter (surname), a su ...
showed that Sharkovskii's theorem holds for triangular mappings, i.e., mappings for which the component depends only on the first components .


References


External links

* * * * {{cite journal, last = Misiurewicz, given = Michal , title = Remarks on Sharkovsky's Theorem, publisher=
The American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...
,Vol. 104, No. 9 (Nov., 1997), pp. 846-847 * Keith Burns and Boris Hasselblatt
The Sharkovsky theorem: a natural direct proof

scholarpedia: Sharkovsky ordering by Aleksandr Nikolayevich Sharkovsky
Theorems in dynamical systems Soviet inventions