Self-similar sets with single-point intersections violating OSC
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Abstract
In the one-dimensional Euclidean space, it was shown that there exists a self-similar set with a single-point intersection that does not satisfy the open set condition (OSC). In the present paper, we prove the existence of a self-similar set with analogous properties in which all mappings are of the form $S_i(x)=q_ix+a_i$ with $q_i>0$.
References
V. Aseev, K. Kamalutdinov, A.Tetenov, General Position Theorem and Its Applications. In: Devaney, R.L., Chan, K.C.,
Vinod Kumar, P. (eds) Topological Dynamics and Topological Data Analysis. IWCTA 2018. Springer Proceedings in
Mathematics & Statistics, vol 350. Springer, Singapore.
G. Edgar Measure, Topology, and Fractal Geometry, 2nd edition. New York: Springer-Verlag, (2008).
J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. 30:5 (1981) 713–747.
Kamalutdinov K. and Tetenov A.Twofold Cantor sets in , Siberian Electr. Math. Rep. 15 (2018), 801–814.
P.A.P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc., 42 (1946), 15–23.
A. Tetenov, K. Kamalutdinov, D. Vaulin Self-similar Jordan arcs which do not satisfy OSC, preprint: arXiv1512.00290,
(2015).
A. Tetenov, K. Kamalutdinov, On one-point intersection property for self-similar fractals, Nonlinearity. 33 (2020),
–415.
M. P. W. Zerner, Weak separation properties for self-similar sets, Proc. Amer. Math. Soc. 124,11 (1996), 3529–3539.
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