# 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 & Continuous Dynamical Systems - A, 2013, 33 (9) : 4305-4321. doi: 10.3934/dcds.2013.33.4305
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