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## Periodic orbits of piecewise monotone maps

Indiana University-Purdue University Indianapolis (IUPUI) / Much is known about periodic orbits in dynamical systems of

continuous interval maps. Of note is the theorem of Sharkovsky.

In 1964 he proved that, for a continuous map $f$ on $\mathbb{R}$,

the existence of periodic orbits of certain periods force the

existence of periodic orbits of certain other periods. Unfortunately

there is currently no analogue of this theorem for maps of $\mathbb{R}$

which are not continuous. Here we consider discontinuous interval maps

of a particular variety, namely piecewise monotone interval maps.

We observe how the presence of a given periodic orbit forces

other periodic orbits, as well as the direct

analogue of Sharkovsky's theorem in special families of

piecewise monotone maps. We conclude by investigating the entropy of

piecewise linear maps. Among particular one parameter families of

piecewise linear maps, entropy remains constant even as the parameter varies.

We provide a simple geometric explanation of this phenomenon known as

entropy locking.

Identifer | oai:union.ndltd.org:IUPUI/oai:scholarworks.iupui.edu:1805/15953 |

Date | 23 April 2018 |

Creators | Cosper, David |

Contributors | Misiurewicz, Michal |

Source Sets | Indiana University-Purdue University Indianapolis |

Language | en_US |

Detected Language | English |

Type | Thesis |

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