#### Event Title

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.