# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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, American Mathematical Society, Providence, RI, 1993. [2] G. Atkinson, A class of transitive cylinder transformations, J. London Math. Soc. (2), 17 (1978), 263-270. [3] A. H. Forrest, The limit points of Weyl sums and other continuous cocycles, J. London Math. Soc. (2), 54 (1996), 440-452. doi: 10.1112/jlms/54.3.440. [4] H. Fujita and D. Shakhmatov, A characterization of compactly generated metric groups, Proc. AMS, 131 (2002), 953-961. doi: 10.1090/S0002-9939-02-06736-9. [5] H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601. [6] W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics," American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955. [7] G. Greschonig and U. Haböck, Nilpotent extensions of minimal homeomorphisms, Ergodic Theory and Dynamical Systems, 25 (2005), 1829-1845. doi: 10.1017/S0143385705000076. [8] K. Kuratowski, "Topology. Vol. II," New edition, revised and augmented, Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. [9] M. Lemańczyk and M. Mentzen, Topological ergodicity of real cocycles over minimal rotations, Monatsh. Math., 134 (2002), 227-246. doi: 10.1007/s605-002-8259-6. [10] M. Mentzen, On groups of essential values of topological cylinder cocycles over minimal rotations, Colloq. Math., 95 (2003), 241-253. doi: 10.4064/cm95-2-8. [11] M. Mentzen, Some applications of groups of essential values of cocycles in topological dynamics, Topol. Methods Nonlinear Anal., 23 (2004), 357-375. [12] S. A. Morris, "Pontryagin Duality and the Structure of Locally Compact Abelian Groups," London Mathematical Society Lecture Note Series, No. 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977. [13] K. Schmidt, "Cocycles on Ergodic Transformation Groups," Macmillan Lectures in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, 1977. [14] K. Schmidt, On recurrence, Z. Wahrsch. Verw. Gebiete, 68 (1984), 75-95. doi: 10.1007/BF00535175.

show all references

##### References:
 [1] E. Akin, "The General Topology of Dynamical Systems," Graduate Studies in Mathematics, 1, American Mathematical Society, Providence, RI, 1993. [2] G. Atkinson, A class of transitive cylinder transformations, J. London Math. Soc. (2), 17 (1978), 263-270. [3] A. H. Forrest, The limit points of Weyl sums and other continuous cocycles, J. London Math. Soc. (2), 54 (1996), 440-452. doi: 10.1112/jlms/54.3.440. [4] H. Fujita and D. Shakhmatov, A characterization of compactly generated metric groups, Proc. AMS, 131 (2002), 953-961. doi: 10.1090/S0002-9939-02-06736-9. [5] H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601. [6] W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics," American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955. [7] G. Greschonig and U. Haböck, Nilpotent extensions of minimal homeomorphisms, Ergodic Theory and Dynamical Systems, 25 (2005), 1829-1845. doi: 10.1017/S0143385705000076. [8] K. Kuratowski, "Topology. Vol. II," New edition, revised and augmented, Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. [9] M. Lemańczyk and M. Mentzen, Topological ergodicity of real cocycles over minimal rotations, Monatsh. Math., 134 (2002), 227-246. doi: 10.1007/s605-002-8259-6. [10] M. Mentzen, On groups of essential values of topological cylinder cocycles over minimal rotations, Colloq. Math., 95 (2003), 241-253. doi: 10.4064/cm95-2-8. [11] M. Mentzen, Some applications of groups of essential values of cocycles in topological dynamics, Topol. Methods Nonlinear Anal., 23 (2004), 357-375. [12] S. A. Morris, "Pontryagin Duality and the Structure of Locally Compact Abelian Groups," London Mathematical Society Lecture Note Series, No. 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977. [13] K. Schmidt, "Cocycles on Ergodic Transformation Groups," Macmillan Lectures in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, 1977. [14] K. Schmidt, On recurrence, Z. Wahrsch. Verw. Gebiete, 68 (1984), 75-95. doi: 10.1007/BF00535175.
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