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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.


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


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


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


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


    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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