Metric cycles, curves and solenoids

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Remarks on multi-marginal symmetric Monge-Kantorovich problems

April 2014, 34(4): 1443-1463. doi: 10.3934/dcds.2014.34.1443

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, Italy
2.	St.Petersburg Branch of the Steklov Mathematical Institute, of the Russian Academy of Sciences, Fontanka 27, 191023 St.Petersburg, Russian Federation

Received November 2012 Revised May 2013 Published October 2013

Abstract
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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.

Keywords: Ambrosio-Kirchheim currents, solenoids., metric currents, Lipschitz curves, Normal currents.

Mathematics Subject Classification: Primary: 49Q15; Secondary: 53C2.

Citation: Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1443-1463. doi: 10.3934/dcds.2014.34.1443

References:

[1]	L. Ambrosio and B. Kirchheim, Currents in metric spaces,, Acta Math., 185 (2000), 1. doi: 10.1007/BF02392711.
[2]	L. Ambrosio and P. Tilli, "Topics on Analysis in Metric Spaces,'', Oxford Lecture Series in Mathematics and its Applications, 25 (2004).
[3]	V. Bangert, Minimal measures and minimizing closed normal one-currents,, Geom. Funct. Anal., 9 (1999), 413. doi: 10.1007/s000390050093.
[4]	V. Bogachev, "Measure Theory. Vol. I, II,", Springer-Verlag, (2007). doi: 10.1007/978-3-540-34514-5.
[5]	L. De Pascale, M. S. Gelli and L. Granieri, Minimal measures, one-dimensional currents and the Monge-Kantorovich problem,, Calc. Var. Partial Differential Equations, 27 (2006), 1. doi: 10.1007/s00526-006-0017-1.
[6]	M. B. Dubashinskiĭ, On uniform approximation by harmonic and almost harmonic vector fields,, (in Russian), 389 (2011), 58. doi: 10.1007/s10958-012-0766-7.
[7]	V. Muñoz and R. Pérez Marco, Ergodic solenoidal homology. II. Density of ergodic solenoids,, Aust. J. Math. Anal. Appl., 6 (2009), 1.
[8]	V. Muñoz and R. Pérez Marco, Schwartzman cycles and ergodic solenoids,, preprint, (2009).
[9]	V. Muñoz and R. Pérez-Marco, Ergodic solenoidal homology: Realization theorem,, Comm. Math. Phys., 302 (2011), 737. doi: 10.1007/s00220-010-1183-8.
[10]	V. Muñoz and R. Pérez Marco, Ergodic solenoids and generalized currents,, Rev. Mat. Complut., 24 (2011), 493. doi: 10.1007/s13163-010-0050-7.
[11]	E. Paolini and E. Stepanov, Decomposition of acyclic normal currents in a metric space,, J. Funct. Anal., 263 (2012), 3358. doi: 10.1016/j.jfa.2012.08.009.
[12]	E. Paolini and E. Stepanov, Structure of metric cycles and normal one-dimensional currents,, J. Funct. Anal., 264 (2013), 1269. doi: 10.1016/j.jfa.2012.12.007.
[13]	S. Schwartzman, Asymptotic cycles,, Ann. of Math. (2), 66 (1957), 270. doi: 10.2307/1969999.
[14]	S. Schwartzman, Asymptotic cycles on non-compact spaces,, Bull. London Math. Soc., 29 (1997), 350. doi: 10.1112/S0024609396002561.
[15]	S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional flows,, St. Petersburg Math. J., 5 (1994), 841.

show all references

References:

[1]	L. Ambrosio and B. Kirchheim, Currents in metric spaces,, Acta Math., 185 (2000), 1. doi: 10.1007/BF02392711.
[2]	L. Ambrosio and P. Tilli, "Topics on Analysis in Metric Spaces,'', Oxford Lecture Series in Mathematics and its Applications, 25 (2004).
[3]	V. Bangert, Minimal measures and minimizing closed normal one-currents,, Geom. Funct. Anal., 9 (1999), 413. doi: 10.1007/s000390050093.
[4]	V. Bogachev, "Measure Theory. Vol. I, II,", Springer-Verlag, (2007). doi: 10.1007/978-3-540-34514-5.
[5]	L. De Pascale, M. S. Gelli and L. Granieri, Minimal measures, one-dimensional currents and the Monge-Kantorovich problem,, Calc. Var. Partial Differential Equations, 27 (2006), 1. doi: 10.1007/s00526-006-0017-1.
[6]	M. B. Dubashinskiĭ, On uniform approximation by harmonic and almost harmonic vector fields,, (in Russian), 389 (2011), 58. doi: 10.1007/s10958-012-0766-7.
[7]	V. Muñoz and R. Pérez Marco, Ergodic solenoidal homology. II. Density of ergodic solenoids,, Aust. J. Math. Anal. Appl., 6 (2009), 1.
[8]	V. Muñoz and R. Pérez Marco, Schwartzman cycles and ergodic solenoids,, preprint, (2009).
[9]	V. Muñoz and R. Pérez-Marco, Ergodic solenoidal homology: Realization theorem,, Comm. Math. Phys., 302 (2011), 737. doi: 10.1007/s00220-010-1183-8.
[10]	V. Muñoz and R. Pérez Marco, Ergodic solenoids and generalized currents,, Rev. Mat. Complut., 24 (2011), 493. doi: 10.1007/s13163-010-0050-7.
[11]	E. Paolini and E. Stepanov, Decomposition of acyclic normal currents in a metric space,, J. Funct. Anal., 263 (2012), 3358. doi: 10.1016/j.jfa.2012.08.009.
[12]	E. Paolini and E. Stepanov, Structure of metric cycles and normal one-dimensional currents,, J. Funct. Anal., 264 (2013), 1269. doi: 10.1016/j.jfa.2012.12.007.
[13]	S. Schwartzman, Asymptotic cycles,, Ann. of Math. (2), 66 (1957), 270. doi: 10.2307/1969999.
[14]	S. Schwartzman, Asymptotic cycles on non-compact spaces,, Bull. London Math. Soc., 29 (1997), 350. doi: 10.1112/S0024609396002561.
[15]	S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional flows,, St. Petersburg Math. J., 5 (1994), 841.

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