The Converse Sharkovskii Theorem and Characterization of Maximal Orbits
Session Number
Project ID: MATH 03
Advisor(s)
Dr. Keith Burns; Northwestern University
Discipline
Mathematics
Start Date
19-4-2023 8:50 AM
End Date
19-4-2023 9:05 AM
Abstract
The Sharkovsky ordering of the natural numbers is 3 ≻ 5 ≻ 7 ≻ ... ≻ 2·3 ≻ 2·5 ≻ 2·7 ≻ ...... ≻ 22·3 ≻ 22·5 ≻ 22·7 ≻ ... ≻ 23 ≻ 22 ≻ 2 ≻ 1. Sharkovsky proved that if m comes before n in this ordering and a continuous map f: R → R has a periodic point with least period m, then it also has a point of least period n. A periodic point for such a map f is called Sharkovsky maximal if no other periodic point of f has a least period that comes earlier in the Sharkovsky ordering than that of p. In the light of Sharkovsky’s Theorem, it is interesting to characterize the Sharkovsky maximal orbits. This has already been done by a number of authors. We do it again. In contrast with previous treatments our arguments are simple. We inherit this approach from Professor Keith Burns from Northwestern University and Professor Boris Hasselblatt from Tufts University.
The Converse Sharkovskii Theorem and Characterization of Maximal Orbits
The Sharkovsky ordering of the natural numbers is 3 ≻ 5 ≻ 7 ≻ ... ≻ 2·3 ≻ 2·5 ≻ 2·7 ≻ ...... ≻ 22·3 ≻ 22·5 ≻ 22·7 ≻ ... ≻ 23 ≻ 22 ≻ 2 ≻ 1. Sharkovsky proved that if m comes before n in this ordering and a continuous map f: R → R has a periodic point with least period m, then it also has a point of least period n. A periodic point for such a map f is called Sharkovsky maximal if no other periodic point of f has a least period that comes earlier in the Sharkovsky ordering than that of p. In the light of Sharkovsky’s Theorem, it is interesting to characterize the Sharkovsky maximal orbits. This has already been done by a number of authors. We do it again. In contrast with previous treatments our arguments are simple. We inherit this approach from Professor Keith Burns from Northwestern University and Professor Boris Hasselblatt from Tufts University.