# American Institute of Mathematical Sciences

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

## Metric cycles, curves and solenoids

 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

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.
Citation: Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete and Continuous Dynamical Systems, 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), 1-80. 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, Oxford University Press, Oxford, 2004. [3] V. Bangert, Minimal measures and minimizing closed normal one-currents, Geom. Funct. Anal., 9 (1999), 413-427. doi: 10.1007/s000390050093. [4] V. Bogachev, "Measure Theory. Vol. I, II," Springer-Verlag, Berlin, 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), 1-23. doi: 10.1007/s00526-006-0017-1. [6] M. B. Dubashinskiĭ, On uniform approximation by harmonic and almost harmonic vector fields, (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 389 (2011), Issledovaniya po Lineinym Operatoram i Teorii Funktsii., 38 (2011), 58-84. 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. [8] V. Muñoz and R. Pérez Marco, Schwartzman cycles and ergodic solenoids, preprint, arXiv:0910.2837, (2009). [9] V. Muñoz and R. Pérez-Marco, Ergodic solenoidal homology: Realization theorem, Comm. Math. Phys., 302 (2011), 737-753. doi: 10.1007/s00220-010-1183-8. [10] V. Muñoz and R. Pérez Marco, Ergodic solenoids and generalized currents, Rev. Mat. Complut., 24 (2011), 493-525. 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), 3358-3390. 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), 1269-1295. doi: 10.1016/j.jfa.2012.12.007. [13] S. Schwartzman, Asymptotic cycles, Ann. of Math. (2), 66 (1957), 270-284. doi: 10.2307/1969999. [14] S. Schwartzman, Asymptotic cycles on non-compact spaces, Bull. London Math. Soc., 29 (1997), 350-352. 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), 841-867.

show all references

##### References:
 [1] L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math., 185 (2000), 1-80. 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, Oxford University Press, Oxford, 2004. [3] V. Bangert, Minimal measures and minimizing closed normal one-currents, Geom. Funct. Anal., 9 (1999), 413-427. doi: 10.1007/s000390050093. [4] V. Bogachev, "Measure Theory. Vol. I, II," Springer-Verlag, Berlin, 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), 1-23. doi: 10.1007/s00526-006-0017-1. [6] M. B. Dubashinskiĭ, On uniform approximation by harmonic and almost harmonic vector fields, (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 389 (2011), Issledovaniya po Lineinym Operatoram i Teorii Funktsii., 38 (2011), 58-84. 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. [8] V. Muñoz and R. Pérez Marco, Schwartzman cycles and ergodic solenoids, preprint, arXiv:0910.2837, (2009). [9] V. Muñoz and R. Pérez-Marco, Ergodic solenoidal homology: Realization theorem, Comm. Math. Phys., 302 (2011), 737-753. doi: 10.1007/s00220-010-1183-8. [10] V. Muñoz and R. Pérez Marco, Ergodic solenoids and generalized currents, Rev. Mat. Complut., 24 (2011), 493-525. 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), 3358-3390. 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), 1269-1295. doi: 10.1016/j.jfa.2012.12.007. [13] S. Schwartzman, Asymptotic cycles, Ann. of Math. (2), 66 (1957), 270-284. doi: 10.2307/1969999. [14] S. Schwartzman, Asymptotic cycles on non-compact spaces, Bull. London Math. Soc., 29 (1997), 350-352. 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), 841-867.
 [1] Mehmet Onur Fen, Marat Akhmet. Impulsive SICNNs with chaotic postsynaptic currents. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1119-1148. doi: 10.3934/dcdsb.2016.21.1119 [2] Luigi Ambrosio, Gianluca Crippa, Philippe G. Lefloch. Leaf superposition property for integer rectifiable currents. Networks and Heterogeneous Media, 2008, 3 (1) : 85-95. doi: 10.3934/nhm.2008.3.85 [3] Didier Bresch, Jacques Simon. Western boundary currents versus vanishing depth. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 469-477. doi: 10.3934/dcdsb.2003.3.469 [4] Reuven Segev, Lior Falach. The co-divergence of vector valued currents. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 687-698. doi: 10.3934/dcdsb.2012.17.687 [5] Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006 [6] Hinke M. Osinga, Arthur Sherman, Krasimira Tsaneva-Atanasova. Cross-currents between biology and mathematics: The codimension of pseudo-plateau bursting. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2853-2877. doi: 10.3934/dcds.2012.32.2853 [7] Katrin Grunert, Helge Holden, Xavier Raynaud. Lipschitz metric for the Camassa--Holm equation on the line. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2809-2827. doi: 10.3934/dcds.2013.33.2809 [8] Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049 [9] Ronald de Man. On composants of solenoids. Electronic Research Announcements, 1995, 1: 87-90. [10] L. Singhal. Cylinder absolute games on solenoids. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2051-2070. doi: 10.3934/dcds.2020352 [11] Manfred Einsiedler and Elon Lindenstrauss. Rigidity properties of \zd-actions on tori and solenoids. Electronic Research Announcements, 2003, 9: 99-110. [12] Francisco J. López-Hernández. Dynamics of induced homeomorphisms of one-dimensional solenoids. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4243-4257. doi: 10.3934/dcds.2018185 [13] Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623 [14] Giuseppe Gaeta, Sebastian Walcher. Higher order normal modes. Journal of Geometric Mechanics, 2020, 12 (3) : 421-434. doi: 10.3934/jgm.2020026 [15] Mohamed Sami ElBialy. Locally Lipschitz perturbations of bisemigroups. Communications on Pure and Applied Analysis, 2010, 9 (2) : 327-349. doi: 10.3934/cpaa.2010.9.327 [16] Alfonso Artigue. Lipschitz perturbations of expansive systems. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1829-1841. doi: 10.3934/dcds.2015.35.1829 [17] Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010 [18] Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54. [19] Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299 [20] Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773

2021 Impact Factor: 1.588

## Metrics

• HTML views (0)
• Cited by (0)

• on AIMS