# American Institute of Mathematical Sciences

2015, 2015(special): 540-548. doi: 10.3934/proc.2015.0540

## Real cocycles of point-distal minimal flows

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

Received  July 2014 Revised  June 2015 Published  November 2015

We generalise the structure theorem for topologically recurrent real skew product extensions of distal minimal compact metric flows in [9] to a class of point distal minimal compact metric flows. While the general case of a point distal flow according to the Veech structure theorem seems hopeless, we prove a result for cocycles of minimal point distal flows without strong Li-Yorke pairs which can be obtained by an almost 1-1 extension of a distal flow with connected fibres. Moreover, a stronger condition on recurrence is necessary. We shall assume that every non-distal point in the point distal compact metric flow is proximal to a point which lifts to recurrent points in the skew product. However, we shall prove that the usual notion of topological recurrence is sufficient for locally connected almost 1-1 extensions of an isometry. This setting includes a well-known example of a point-distal flow by Mary Rees.
Citation: Gernot Greschonig. Real cocycles of point-distal minimal flows. Conference Publications, 2015, 2015 (special) : 540-548. doi: 10.3934/proc.2015.0540
##### References:
 [1] E. Akin and E. Glasner, Topological ergodic decomposition and homogeneous flows, Topological Dynamics and Applications (eds. M. G. Nerurkar, D. P. Dokken and D. B. Ellis), Contemp. Math., 215, AMS, Providence, RI, (1998), 43-52. [2] G. Atkinson, A class of transitive cylinder transformations, J. London Math. Soc. (2), 17 (1978), 263-270. [3] R. Ellis, Distal transformation groups, Pacific Journal Math., 8 (1958), 401-405. [4] R. Ellis, The Veech structure theorem, Trans. Amer. Math. Soc., 186 (1973), 203-218. [5] E. E. Floyd, A nonhomogeneous minimal set, Bull. Amer. Math. Soc., 55 (1949), 957-960. [6] H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. [7] E. Glasner, Relatively invariant measures, Pacific J. Math., 58 (1975), 393-410. [8] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, American Mathematical Society Colloquium Publications, Vol. 36, AMS, Providence, R. I., 1955. [9] G. Greschonig, Real extensions of distal minimal flows and continuous topological ergodic decompositions, Proc. Lond. Math. Soc. (3), 109 (2014), 213-240. [10] K. N. Haddad and A. S. A. Johnson, Auslander systems, Proc. Amer. Math. Soc., 125 (1997), 2161-2170. [11] M. Lemańczyk and M. Mentzen, Topological ergodicity of real cocycles over minimal rotations, Monatsh. Math., 134 (2002), 227-246. [12] D. McMahon and T. S. Wu, On the connectedness of homomorphisms in topological dynamics, Trans. Amer. Math. Soc., 217 (1976), 257-270. [13] M. Rees, A point distal transformation of the torus, Israel J. Math., 32 (1979), 201-208. [14] K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, 1977. [15] W. A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205-242.

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##### References:
 [1] E. Akin and E. Glasner, Topological ergodic decomposition and homogeneous flows, Topological Dynamics and Applications (eds. M. G. Nerurkar, D. P. Dokken and D. B. Ellis), Contemp. Math., 215, AMS, Providence, RI, (1998), 43-52. [2] G. Atkinson, A class of transitive cylinder transformations, J. London Math. Soc. (2), 17 (1978), 263-270. [3] R. Ellis, Distal transformation groups, Pacific Journal Math., 8 (1958), 401-405. [4] R. Ellis, The Veech structure theorem, Trans. Amer. Math. Soc., 186 (1973), 203-218. [5] E. E. Floyd, A nonhomogeneous minimal set, Bull. Amer. Math. Soc., 55 (1949), 957-960. [6] H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. [7] E. Glasner, Relatively invariant measures, Pacific J. Math., 58 (1975), 393-410. [8] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, American Mathematical Society Colloquium Publications, Vol. 36, AMS, Providence, R. I., 1955. [9] G. Greschonig, Real extensions of distal minimal flows and continuous topological ergodic decompositions, Proc. Lond. Math. Soc. (3), 109 (2014), 213-240. [10] K. N. Haddad and A. S. A. Johnson, Auslander systems, Proc. Amer. Math. Soc., 125 (1997), 2161-2170. [11] M. Lemańczyk and M. Mentzen, Topological ergodicity of real cocycles over minimal rotations, Monatsh. Math., 134 (2002), 227-246. [12] D. McMahon and T. S. Wu, On the connectedness of homomorphisms in topological dynamics, Trans. Amer. Math. Soc., 217 (1976), 257-270. [13] M. Rees, A point distal transformation of the torus, Israel J. Math., 32 (1979), 201-208. [14] K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, 1977. [15] W. A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205-242.
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