*Published Paper*

**Inserted:** 3 nov 2012

**Last Updated:** 12 oct 2019

**Journal:** Discrete Contin. Dyn. Syst.

**Volume:** 34

**Number:** 4

**Pages:** 1443–1463

**Year:** 2014

**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 Ces\`{a}ro 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 Ces\`{a}ro means of this curve over every sequence of symmetric bounded intervals, that is, in other words, such a cycle is a solenoid.

**Keywords:**
metric current, Lipschitz curve, asymptotic cycle, solenoid

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