Zulip Chat Archive

Stream: general

Topic: Sharkovsky's Theorem


view this post on Zulip Kenny Lau (Mar 28 2019 at 15:03):

http://mathworld.wolfram.com/SharkovskysTheorem.html

Order the natural numbers as such: 3<5<7<9<11<13<....<6<10<14<18<22<26<....<12<20<28<36<44<52<...<8<4<2<1
Now let p<q. If f:R->R is continuous and there is a point of least period p then there is a point of least period q.

(period here referring to f(f(f(...f(x)...))))

view this post on Zulip Kevin Buzzard (Mar 28 2019 at 15:04):

Isn't that just the best theorem?

view this post on Zulip Kevin Buzzard (Mar 28 2019 at 15:04):

For me what is most remarkable about it is that any result of that nature should even be true. If there is a point of period 3 then there is a point of period n for all n?? Really? Yes!

view this post on Zulip Sebastien Gouezel (Mar 28 2019 at 15:25):

The proof is not even hard. Just needs the intermediate value theorem, and a careful study of overlaps of intervals.


Last updated: May 12 2021 at 05:19 UTC