### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

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

**Metric cycles, curves and solenoids.** / Gueorguiev, Vladimir Simeonov; Stepanov, Eugene.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems- Series A*, vol. 34, no. 4, pp. 1443-1463. https://doi.org/10.3934/dcds.2014.34.1443

}

TY - JOUR

T1 - Metric cycles, curves and solenoids

AU - Gueorguiev, Vladimir Simeonov

AU - Stepanov, Eugene

PY - 2014/4

Y1 - 2014/4

N2 - 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.

AB - 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.

KW - Ambrosio-kirchheim currents

KW - Lipschitz curves

KW - Metric currents

KW - Normal currents

KW - Solenoids

UR - http://www.scopus.com/inward/record.url?scp=84886438502&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84886438502&partnerID=8YFLogxK

U2 - 10.3934/dcds.2014.34.1443

DO - 10.3934/dcds.2014.34.1443

M3 - Article

VL - 34

SP - 1443

EP - 1463

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

SN - 1078-0947

IS - 4

ER -