In
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