### Abstract

We look at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f 2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.).

Original language | English |
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Pages (from-to) | 773-780 |

Number of pages | 8 |

Journal | American Mathematical Monthly |

Volume | 122 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2015 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On devaney chaos and dense periodic points : Period 3 and Higher implies chaos.** / Che Dzul-Kifli, Syahida; Good, Chris.

Research output: Contribution to journal › Article

*American Mathematical Monthly*, vol. 122, no. 8, pp. 773-780. https://doi.org/10.4169/amer.math.monthly.122.8.773

}

TY - JOUR

T1 - On devaney chaos and dense periodic points

T2 - Period 3 and Higher implies chaos

AU - Che Dzul-Kifli, Syahida

AU - Good, Chris

PY - 2015

Y1 - 2015

N2 - We look at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f 2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.).

AB - We look at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f 2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.).

UR - http://www.scopus.com/inward/record.url?scp=84975462816&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975462816&partnerID=8YFLogxK

U2 - 10.4169/amer.math.monthly.122.8.773

DO - 10.4169/amer.math.monthly.122.8.773

M3 - Article

AN - SCOPUS:84975462816

VL - 122

SP - 773

EP - 780

JO - American Mathematical Monthly

JF - American Mathematical Monthly

SN - 0002-9890

IS - 8

ER -