Sharkovskii's theorem

Sharkovskii's theorem   (Redirected from Sarkovskii's theorem) Jump to navigation Jump to search In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about discrete dynamical systems.[1] One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.

Contents 1 Statement 2 Generalizations and related results 3 References 4 External links Statement For some interval {displaystyle Isubset mathbb {R} } , suppose that {displaystyle f:Ito I} is a continuous function. The number {displaystyle x} is called a periodic point of period {displaystyle m} if {displaystyle f^{(m)}(x)=x} , where {displaystyle f^{(m)}} denotes the iterated function obtained by composition of {displaystyle m} copies of {displaystyle f} . The number {displaystyle x} is said to have least period {displaystyle m} if, in addition, {displaystyle f^{(k)}(x)neq x} for all {displaystyle 0

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