Calculus of Variations and Geometric Measure Theory
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V. Georgiev - E. Stepanov

Metric cycles, curves and solenoids

created by stepanov on 03 Nov 2012
modified on 11 Oct 2014

[BibTeX]

Published Paper

Inserted: 3 nov 2012
Last Updated: 11 oct 2014

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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