Metric cycles, curves and solenoids

Vladimir Georgiev, Eugene Stepanov

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 languageEnglish
Pages (from-to)1443-1463
Number of pages21
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume34
Issue number4
DOIs
Publication statusPublished - 2014 Apr 1
Externally publishedYes

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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