September  2013, 33(9): 4305-4321. doi: 10.3934/dcds.2013.33.4305

Regularity of topological cocycles of a class of non-isometric minimal homeomorphisms

1. 

Institute of Discrete Mathematics and Geometry, Vienna University of Technology (TU Vienna), Wiedner Hauptstraße 8-10, A1040 Vienna, Austria

Received  December 2010 Revised  April 2011 Published  March 2013

We study topological cocycles of a class of non-isometric distal minimal homeomorphisms of multidimensional tori, introduced by Furstenberg in [5] as iterated skew product extensions by the torus, starting with an irrational rotation. We prove that there are no topological type ${III}_0$ cocycles of these homeomorphisms with values in an Abelian locally compact group. Moreover, under the assumption that the Abelian locally compact group has no non-trivial connected compact subgroup, we show that a topologically recurrent cocycle is always regular, i.e. it is topologically cohomologous to a cocycle with values only in the essential range. These properties are well-known for topological cocycles of minimal rotations on compact metric groups (cf. [6], [2], [9], and [10]), but the distal minimal homeomorphisms considered in this paper are far from the isometric behaviour of minimal rotations and do not admit rigidity times.
Citation: Gernot Greschonig. Regularity of topological cocycles of a class of non-isometric minimal homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4305-4321. doi: 10.3934/dcds.2013.33.4305
References:
[1]

E. Akin, "The General Topology of Dynamical Systems,", Graduate Studies in Mathematics, 1 (1993).   Google Scholar

[2]

G. Atkinson, A class of transitive cylinder transformations,, J. London Math. Soc. (2), 17 (1978), 263.   Google Scholar

[3]

A. H. Forrest, The limit points of Weyl sums and other continuous cocycles,, J. London Math. Soc. (2), 54 (1996), 440.  doi: 10.1112/jlms/54.3.440.  Google Scholar

[4]

H. Fujita and D. Shakhmatov, A characterization of compactly generated metric groups,, Proc. AMS, 131 (2002), 953.  doi: 10.1090/S0002-9939-02-06736-9.  Google Scholar

[5]

H. Furstenberg, Strict ergodicity and transformation of the torus,, Amer. J. Math., 83 (1961), 573.   Google Scholar

[6]

W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics,", American Mathematical Society Colloquium Publications, (1955).   Google Scholar

[7]

G. Greschonig and U. Haböck, Nilpotent extensions of minimal homeomorphisms,, Ergodic Theory and Dynamical Systems, 25 (2005), 1829.  doi: 10.1017/S0143385705000076.  Google Scholar

[8]

K. Kuratowski, "Topology. Vol. II,", New edition, (1968).   Google Scholar

[9]

M. Lemańczyk and M. Mentzen, Topological ergodicity of real cocycles over minimal rotations,, Monatsh. Math., 134 (2002), 227.  doi: 10.1007/s605-002-8259-6.  Google Scholar

[10]

M. Mentzen, On groups of essential values of topological cylinder cocycles over minimal rotations,, Colloq. Math., 95 (2003), 241.  doi: 10.4064/cm95-2-8.  Google Scholar

[11]

M. Mentzen, Some applications of groups of essential values of cocycles in topological dynamics,, Topol. Methods Nonlinear Anal., 23 (2004), 357.   Google Scholar

[12]

S. A. Morris, "Pontryagin Duality and the Structure of Locally Compact Abelian Groups,", London Mathematical Society Lecture Note Series, 29 (1977).   Google Scholar

[13]

K. Schmidt, "Cocycles on Ergodic Transformation Groups,", Macmillan Lectures in Mathematics, 1 (1977).   Google Scholar

[14]

K. Schmidt, On recurrence,, Z. Wahrsch. Verw. Gebiete, 68 (1984), 75.  doi: 10.1007/BF00535175.  Google Scholar

show all references

References:
[1]

E. Akin, "The General Topology of Dynamical Systems,", Graduate Studies in Mathematics, 1 (1993).   Google Scholar

[2]

G. Atkinson, A class of transitive cylinder transformations,, J. London Math. Soc. (2), 17 (1978), 263.   Google Scholar

[3]

A. H. Forrest, The limit points of Weyl sums and other continuous cocycles,, J. London Math. Soc. (2), 54 (1996), 440.  doi: 10.1112/jlms/54.3.440.  Google Scholar

[4]

H. Fujita and D. Shakhmatov, A characterization of compactly generated metric groups,, Proc. AMS, 131 (2002), 953.  doi: 10.1090/S0002-9939-02-06736-9.  Google Scholar

[5]

H. Furstenberg, Strict ergodicity and transformation of the torus,, Amer. J. Math., 83 (1961), 573.   Google Scholar

[6]

W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics,", American Mathematical Society Colloquium Publications, (1955).   Google Scholar

[7]

G. Greschonig and U. Haböck, Nilpotent extensions of minimal homeomorphisms,, Ergodic Theory and Dynamical Systems, 25 (2005), 1829.  doi: 10.1017/S0143385705000076.  Google Scholar

[8]

K. Kuratowski, "Topology. Vol. II,", New edition, (1968).   Google Scholar

[9]

M. Lemańczyk and M. Mentzen, Topological ergodicity of real cocycles over minimal rotations,, Monatsh. Math., 134 (2002), 227.  doi: 10.1007/s605-002-8259-6.  Google Scholar

[10]

M. Mentzen, On groups of essential values of topological cylinder cocycles over minimal rotations,, Colloq. Math., 95 (2003), 241.  doi: 10.4064/cm95-2-8.  Google Scholar

[11]

M. Mentzen, Some applications of groups of essential values of cocycles in topological dynamics,, Topol. Methods Nonlinear Anal., 23 (2004), 357.   Google Scholar

[12]

S. A. Morris, "Pontryagin Duality and the Structure of Locally Compact Abelian Groups,", London Mathematical Society Lecture Note Series, 29 (1977).   Google Scholar

[13]

K. Schmidt, "Cocycles on Ergodic Transformation Groups,", Macmillan Lectures in Mathematics, 1 (1977).   Google Scholar

[14]

K. Schmidt, On recurrence,, Z. Wahrsch. Verw. Gebiete, 68 (1984), 75.  doi: 10.1007/BF00535175.  Google Scholar

[1]

Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure & Applied Analysis, 2021, 20 (1) : 193-214. doi: 10.3934/cpaa.2020263

[2]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020350

[3]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[4]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[5]

Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]