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

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