### Abstract

We prove that every one-dimensional real Ambrosio-Kirchheim current with zero boundary (i.e. a cycle) in a lot of reasonable spaces (including all finite-dimensional normed spaces) can be represented by a Lipschitz curve parameterized over the real line through a suitable limit of Cesàro means of this curve over a subsequence of symmetric bounded intervals (viewed as currents). It is further shown that in such spaces, if a cycle is indecomposable, i.e. does not contain "nontrivial" subcycles, then it can be represented again by a Lipschitz curve parameterized over the real line through a limit of Cesàro means of this curve over every sequence of symmetric bounded intervals, that is, in other words, such a cycle is a solenoid.

Original language | English |
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Pages (from-to) | 1443-1463 |

Number of pages | 21 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 34 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2014 Apr 1 |

Externally published | Yes |

### Keywords

- Ambrosio-kirchheim currents
- Lipschitz curves
- Metric currents
- Normal currents
- Solenoids

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*34*(4), 1443-1463. https://doi.org/10.3934/dcds.2014.34.1443