### Abstract

We show that under suitable assumptions, Poincaré recurrences of a dynamical system determine its topology in phase space. Therefore, dynamical systems with the same recurrences are dynamically equivalent. This conclusion can be drawn from a theorem proved in this paper which states that the recurrence matrix determines the topology of closed sets. The theorem states that if a set of points M is mapped onto another set N, such that two points in N are closer than some prescribed fixed distance if and only if the corresponding points in M are closer than some, in general different, prescribed fixed distance, then both sets are homeomorphic, i.e., identical up to a continuous change in the coordinate system. The theorem justifies a range of methods in nonlinear dynamics which are based on recurrence properties.

Original language | English |
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Article number | 023104 |

Number of pages | 5 |

Journal | Chaos |

Volume | 19 |

Issue number | 2 |

Early online date | 4 May 2009 |

DOIs | |

Publication status | Published - Jun 2009 |

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## Cite this

Robinson, G., & Thiel, M. (2009). Recurrences determine the dynamics.

*Chaos*,*19*(2), [023104]. https://doi.org/10.1063/1.3117151