Metric cycles, curves and solenoids

Vladimir Georgiev; Eugene Stepanov

doi:10.3934/dcds.2014.34.1443

Article Contents

2014, Volume 34, Issue 4: 1443-1463. Doi: 10.3934/dcds.2014.34.1443

This issue Previous Article Optimal transport and large number of particles Next Article Remarks on multi-marginal symmetric Monge-Kantorovich problems

Metric cycles, curves and solenoids

Vladimir Georgiev^1, and
Eugene Stepanov^2,

1.
Dipartimento di Matematica "L. Tonelli", Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa
2.
St.Petersburg Branch of the Steklov Mathematical Institute, of the Russian Academy of Sciences, Fontanka 27, 191023 St.Petersburg

Received: October 31, 2012

Revised: April 30, 2013

Published: March 2014

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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.

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Mathematics Subject Classification: Primary: 49Q15; Secondary: 53C23.

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