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, (1998), 43. Google Scholar

[2]

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

[3]

R. Ellis, Distal transformation groups,, Pacific Journal Math., 8 (1958), 401. Google Scholar

[4]

R. Ellis, The Veech structure theorem,, Trans. Amer. Math. Soc., 186 (1973), 203. Google Scholar

[5]

E. E. Floyd, A nonhomogeneous minimal set,, Bull. Amer. Math. Soc., 55 (1949), 957. Google Scholar

[6]

H. Furstenberg, The structure of distal flows,, Amer. J. Math., 85 (1963), 477. Google Scholar

[7]

E. Glasner, Relatively invariant measures,, Pacific J. Math., 58 (1975), 393. Google Scholar

[8]

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

[9]

G. Greschonig, Real extensions of distal minimal flows and continuous topological ergodic decompositions,, Proc. Lond. Math. Soc. (3), 109 (2014), 213. Google Scholar

[10]

K. N. Haddad and A. S. A. Johnson, Auslander systems,, Proc. Amer. Math. Soc., 125 (1997), 2161. Google Scholar

[11]

M. Lemańczyk and M. Mentzen, Topological ergodicity of real cocycles over minimal rotations,, Monatsh. Math., 134 (2002), 227. Google Scholar

[12]

D. McMahon and T. S. Wu, On the connectedness of homomorphisms in topological dynamics,, Trans. Amer. Math. Soc., 217 (1976), 257. Google Scholar

[13]

M. Rees, A point distal transformation of the torus,, Israel J. Math., 32 (1979), 201. Google Scholar

[14]

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

[15]

W. A. Veech, Point-distal flows,, Amer. J. Math., 92 (1970), 205. Google Scholar

show all references

References:
[1]

E. Akin and E. Glasner, Topological ergodic decomposition and homogeneous flows,, Topological Dynamics and Applications (eds. M. G. Nerurkar, (1998), 43. Google Scholar

[2]

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

[3]

R. Ellis, Distal transformation groups,, Pacific Journal Math., 8 (1958), 401. Google Scholar

[4]

R. Ellis, The Veech structure theorem,, Trans. Amer. Math. Soc., 186 (1973), 203. Google Scholar

[5]

E. E. Floyd, A nonhomogeneous minimal set,, Bull. Amer. Math. Soc., 55 (1949), 957. Google Scholar

[6]

H. Furstenberg, The structure of distal flows,, Amer. J. Math., 85 (1963), 477. Google Scholar

[7]

E. Glasner, Relatively invariant measures,, Pacific J. Math., 58 (1975), 393. Google Scholar

[8]

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

[9]

G. Greschonig, Real extensions of distal minimal flows and continuous topological ergodic decompositions,, Proc. Lond. Math. Soc. (3), 109 (2014), 213. Google Scholar

[10]

K. N. Haddad and A. S. A. Johnson, Auslander systems,, Proc. Amer. Math. Soc., 125 (1997), 2161. Google Scholar

[11]

M. Lemańczyk and M. Mentzen, Topological ergodicity of real cocycles over minimal rotations,, Monatsh. Math., 134 (2002), 227. Google Scholar

[12]

D. McMahon and T. S. Wu, On the connectedness of homomorphisms in topological dynamics,, Trans. Amer. Math. Soc., 217 (1976), 257. Google Scholar

[13]

M. Rees, A point distal transformation of the torus,, Israel J. Math., 32 (1979), 201. Google Scholar

[14]

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

[15]

W. A. Veech, Point-distal flows,, Amer. J. Math., 92 (1970), 205. Google Scholar

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