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A Besicovitch cylindrical transformation with Hölder function

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  • For any $\gamma\in(0,1)$ and any $\varepsilon>0$ we construct a cylindrical cascade over some irrational circle rotation with a $\gamma$-Hölder function such that the Besicovitch condition holds and the Hausdorff dimension of the set of points in the circle having discrete orbits is more than $1-\gamma-\varepsilon$. This result gives the answers to some questions of K. Frączek and M. Lemańczyk [1].
    Mathematics Subject Classification: 37B05, 37C45.

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  • [1]

    K. Frączek and M. Lemańczyk, On the Hausdorff dimension of the set of closed orbits for a cylindrical transformation, Nonlinearity, 23 (2010), 2393-2422.doi: 10.1088/0951-7715/23/10/003.

    [2]

    A. S. Besicovitch, A problem on topological transformations of the plane. II, Proc. Cambridge Philos. Soc., 47 (1951), 38-45.doi: 10.1017/S0305004100026347.

    [3]

    W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ., Vol. 36, Amer. Math. Soc., Providence, RI, 1955.

    [4]

    E. Dymek, Transitive cylinder flows whose set of discrete points is of full Hausdorff dimension, arXiv:1303.3099v1, 2013.

    [5]

    A. Kochergin, A mixing special flow over a circle rotation with almost Lipschitz function, Sbornik: Mathematics, 193 (2002), 359-385.doi: 10.1070/SM2002v193n03ABEH000636.

    [6]

    K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003.doi: 10.1002/0470013850.

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