In
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 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
, suppose that
is a
continuous function. The number
is called a ''periodic point of period
'' if
, where
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
copies of
. The number
is said to have ''least period
'' if, in addition,
for all