Article Contents
Article Contents

# A Besicovitch cylindrical transformation with Hölder function

• 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.

 Citation:

•  [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.