Certain Singular Distributions and Fractals

Symon, Serbenyuk

doi:10.2478/tmmp-2021-0026

Abstract

In the presented paper, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in their own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.

References

[1] DE AMO, E.—CARRILLO, M.D. —FERNÁNDEZ-SÁNCHEZ, J.: A Salem generalized function, Acta Math. Hungar. 151 (2017), no. 2, 361–378. https://doi.org/10.1007/s10474-017-0690-x10.1007/s10474-017-0690-x
Search in Google Scholar Back to article
[2] BALANKIN, A.S.—BORY REYES, J.—LUNA-ELIZARRARÁS, M.E. —SHAPIRO, M.: Cantor-type sets in hyperbolic numbers, Fractals 24 (2016), no. 4, Paper no. 1650051.
Search in Google Scholar Back to article
[3] BRODERICK, R.—FISHMAN, L.—REICH, A.: Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler, Mosc. J. Comb. Number Theory 1 (2011), no. 4, 3–12.
Search in Google Scholar Back to article
[4] BUNDE, A.—HAVLIN, S.: Fractals in Science, Springer-Verlag, Berlin, 1994.10.1007/978-3-642-77953-4
Search in Google Scholar Back to article
[5] DANI, S.G.—SHAH, HEMANGI: Badly approximable numbers and vectors in Cantor-like sets, Proc. Amer. Math. Soc. 140 (2012), 2575–2587.10.1090/S0002-9939-2011-11105-5
Search in Google Scholar Back to article
[6] DIMARTINO, R.—URBINA, W. O.: On Cantor-like sets and Cantor-Lebesgue singular functions, https://arxiv.org/pdf/1403.6554.pdf
Search in Google Scholar Back to article
[7] DIMARTINO, R.—URBINA, W. O.: Excursions on Cantor-like Sets, https://arxiv.org/pdf/1411.7110.pdf
Search in Google Scholar Back to article
[8] FALCONER, K.: Techniques in Fractal Geometry, John Willey and Sons, Ltd., Chichester, 1997.10.2307/2533585
Search in Google Scholar Back to article
[9] FALCONER, K.: Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2004.10.1002/0470013850
Search in Google Scholar Back to article
[10] FENG, D. J.: The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers, Adv. Math. 195 (2005), 24–101.10.1016/j.aim.2004.06.011
Search in Google Scholar Back to article
[11] HUA, S.—RAO, H.—WEN, Z. ET AL.: textitOn the structures and dimensions of Moran sets, Sci. China Ser. A-Math. 43 (2000), no. 8, 836–852
Search in Google Scholar Back to article
[12] ITO, S.—SADAHIRO, T.: Beta-expansions with negative bases, Integers 9 (2009), 239–259.10.1515/INTEG.2009.023
Search in Google Scholar Back to article
[13] KÄENMÄKI, A.—LI, B.—SUOMALA, V.: Local dimensions in Moran constructions, Nonlinearity 29 (2016), no. 3, 807–822.
Search in Google Scholar Back to article
[14] KALPAZIDOU, S.—KNOPFMACHER, A.—KNOPFMACHER, J.: Lüroth-type alternating series representations for real numbers, Acta Arithmetica 55 (1990), 311–322.10.4064/aa-55-4-311-322
Search in Google Scholar Back to article
[15] KATSUURA, H.: Continuous nowhere-differentiable functions - an application of contraction mappings, Amer. Math. Monthly 98 (1991), no. 5, 411–416, https://doi.org/10.1080/00029890.1991.1200077810.1080/00029890.1991.12000778
Search in Google Scholar Back to article
[16] KAWAMURA, K.: The derivative of Lebesgue’s singular function, In: Summer Symposium 2010, Real Anal. Exchange, pp. 83–85.
Search in Google Scholar Back to article
[17] KENNEDY, J. A.—YORKE, J. A.: Bizarre topology is natural in dynamical systems, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 3, 309–316.
Search in Google Scholar Back to article
[18] LI, J.— WU, M.: Pointwise dimensions of general Moran measures with open set condition, Sci. China, Math. 54 (2011), 699–710.
Search in Google Scholar Back to article
[19] MANDELBROT, B.: Fractals: Form, Chance and Dimension. W. H. Freeman and Co., San Francisco, Calif. 1977.
Search in Google Scholar Back to article
[20] MANDELBROT, B.: The Fractal Geometry of Nature. 18th printing, Freeman, New York, 1999.
Search in Google Scholar Back to article
[21] MORAN, PA. P.: Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), no. 1, 15–23, doi:10.1017/S0305004100022684.10.1017/S0305004100022684
Search in Google Scholar Back to article
[22] PALIS, J.—TAKENS, F.: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors. In: Cambridge Studies in Advanced Mathematics Vol. 35, Cambridge University Press, Cambridge, 1993.
Search in Google Scholar Back to article
[23] POLLICOTT, M.—SIMON, K.: The Hausdorff dimension of λ-expansions with deleted digits, Trans. Amer. Math. Soc. 347 (1995), no. 3, 967–983. https://doi.org/10.1090/S0002-9947-1995-1290729-010.1090/S0002-9947-1995-1290729-0
Search in Google Scholar Back to article
[24] RÉNYI, A.: Representations for real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hungar. 8 (1957), 477–493.10.1007/BF02020331
Search in Google Scholar Back to article
[25] SALEM, R.: On some singular monotonic functions which are stricly increasing, Trans. Amer. Math. Soc. 53 (1943), 423–439.10.1090/S0002-9947-1943-0007929-6
Search in Google Scholar Back to article
[26] SERBENYUK, S. O.: Topological, metric and fractal properties of one set defined by using the s-adic representation. In: XIV International Scientific Kravchuk Conference: Conference materials II, Kyiv: National Technical University of Ukraine “KPI”, 2012, p.220. (In Ukrainian) https://www.researchgate.net/publication/311665455
Search in Google Scholar Back to article
[27] SERBENYUK, S.O.: Topological, metric and fractal properties of sets of class generated by one set with using the s-adic representation. In: International Conference “Dynamical Systems and their Applications: Abstracts, Kyiv: Institute of Mathematics of NAS of Ukraine, 2012, p. 42. (In Ukrainian) https://www.researchgate.net/publication/311415778
Search in Google Scholar Back to article
[28] SERBENYUK, S. O.: Topological, metric and fractal properties of the set with parameter, that the set defined by s-adic representation of numbers. In: International Conference “Modern Stochastics: Theory and Applications III dedicated to 100th anniversary of B. V. Gnedenko and 80th anniversary of M. I. Yadrenko: Abstracts, Kyiv: Taras Shevchenko National University of Kyiv, 2012, p. 13. https://www.researchgate.net/publication/311415501
Search in Google Scholar Back to article
[29] SERBENYUK, S.O.: Topological, metric, and fractal properties of one set of real numbers such that it defined in terms of the s-adic representation, Naukovyi Chasopys NPU im. M. P. Dragomanova. Seria 1. Phizyko-matematychni Nauky [Trans. Natl. Pedagog. Mykhailo Dragomanov University. Ser. 1. Phys. Math.] 11 (2010), 241–250. (in Ukrainian) https://www.researchgate.net/publication/292606441
Search in Google Scholar Back to article
[30] SERBENYUK, S. O.: Topological, metric properties and using one generalizad set determined by the s-adic representation with a parameter, Naukovyi Chasopys NPU im. M.P. Dragomanova. Ser. 1. Phizyko-matematychni Nauky [Trans. Natl. Pedagog. Mykhailo Dragomanov University. Ser. 1. Phys. Math.] 12 (2011), 66–75. (In Ukrainian) https://www.researchgate.net/publication/292970196
Search in Google Scholar Back to article
[31] SERBENYUK, S. O.: On some sets of real numbers such that defined by nega-s-adic and Cantor nega-s-adic representations, Naukovyi Chasopys NPU im. M. P. Dragomanova. Seria 1. Phizyko-matematychni Nauky [Trans. Natl. Pedagog. Mykhailo Dragomanov Univ. Ser.1. Phys. Math.] 15 (2013), 168–187. (In Ukrainian) https://www.researchgate.net/publication/292970280
Search in Google Scholar Back to article
[32] SERBENYUK, S. O.: Functions, that defined by functional equations systems in terms of Cantor series representation of numbers, Naukovi Zapysky NaUKMA 165 (2015), 34–40. (In Ukrainian) https://www.researchgate.net/publication/292606546
Search in Google Scholar Back to article
[33] SERBENYUK, S.: On some generalizations of real numbers representations, arXiv:1602.07929v1. (In Ukrainian)
Search in Google Scholar Back to article
[34] SERBENYUK, S.: One one class of fractal sets, https://arxiv.org/pdf/1703.05262.pdf
Search in Google Scholar Back to article
[35] SERBENYUK, S.: More on one class of fractals, arXiv:1706.01546v1.
Search in Google Scholar Back to article
[36] SERBENYUK, S.O.: One distribution function on the Moran sets, Azerb. J. Math. 10 (2020), no. 2, 12–30, arXiv:1808.00395v1.
Search in Google Scholar Back to article
[37] SERBENYUK, S.: Nega-Q˜\[\tilde Q\]-representation as a generalization of certain alternating representations of real numbers, Bull. Taras Shevchenko Natl. Univ. Kyiv Math. Mech. 1 (35) (2016), no.1, 32–39. (In Ukrainian) https://www.researchgate.net/publication/308273000
Search in Google Scholar Back to article
[38] SERBENYUK, S.: On one class of functions with complicated local structure, Šiauliai Mathematical Seminar 11 (19) (2016), 75–88.
Search in Google Scholar Back to article
[39] SERBENYUK, S. O.: Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers, Zh. Mat. Fiz. Anal. Geom. 13 (2017), no. 1, 57–81. https://doi.org/10.15407/mag13.01.05710.15407/mag13.01.057
Search in Google Scholar Back to article
[40] SERBENYUK, S. O.: Non-differentiable functions defined in terms of classical representations of real numbers, Zh. Mat. Fiz. Anal. Geom. 14 (2018), no. 2, 197–213. https://doi.org/10.15407/mag14.02.19710.15407/mag14.02.197
Search in Google Scholar Back to article
[41] SERBENYUK, S. O.: Preserving the Hausdorff-Besicovitch dimension by monotonic singular distribution functions. In: Second Interuniversity Scientific Conference on Mathematics and Physics for Young Scientists: Abstracts. Institute of Mathematics of NAS of Ukraine, Kyiv (2011). pp. 106–107. (In Ukrainian) https://www.researchgate.net/publication/301637057
Search in Google Scholar Back to article
[42] SERBENYUK, S.: On one fractal property of the Minkowski function, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Ser. A. Matematicás 112 (2018), no. 2, 555–559, doi:10.1007/s13398-017-0396-5.10.1007/s13398-017-0396-5
Search in Google Scholar Back to article
[43] SERBENYUK, S.: On one application of infinite systems of functional equations in function theory, Tatra Mt. Math. Publ. 74 (2019), 117–144. https://doi.org/10.2478/tmmp-2019-002410.2478/tmmp-2019-0024
Search in Google Scholar Back to article
[44] SERBENYUK, S.: On certain maps defined by infinite sums, J. Anal. 28 (2020), no. 4, 987–1007. https://doi.org/10.1007/s41478-020-00229-x10.1007/s41478-020-00229-x
Search in Google Scholar Back to article
[45] TAYLOR, T. D.—HUDSON, C.— ANDERSON, A.: Examples of using binary Cantor sets to study the connectivity of Sierpinski relatives, Fractals 20 (2012), no. 1, 61–75.
Search in Google Scholar Back to article
[46] TÉLLEZ-SÁNCHEZ, G. Y.—BORY-REYES, J.: More about Cantor like sets in hyperbolic numbers, Fractals 25 (2017), no. 5, Paper no. 1750046.
Search in Google Scholar Back to article
[47] WANG, B. W.— WU, J.: Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math. 218 (2008), 1319–1339.10.1016/j.aim.2008.03.006
Search in Google Scholar Back to article
[48] WIKIPEDIA CONTRIBUTORS: Fractal, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Fractal
Search in Google Scholar Back to article
[49] WIKIPEDIA CONTRIBUTORS: Pathological (mathematics), Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Pathological_(mathematics)
Search in Google Scholar Back to article
[50] WIKIPEDIA CONTRIBUTORS: Self-similarity, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Self-similarity
Search in Google Scholar Back to article
[51] WIKIPEDIA CONTRIBUTORS: Singular function, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Singular_function
Search in Google Scholar Back to article
[52] WIKIPEDIA CONTRIBUTORS: Thomae’s function, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Thomae’s_function
Search in Google Scholar Back to article
[53] WU, J.: On the sum of degrees of digits occurring in continued fraction expansions of Laurent series, Math. Proc. Camb. Philos. Soc. 138 (2005), 9–20.10.1017/S0305004104008163
Search in Google Scholar Back to article
[54] WU, M.: The singularity spectrum f (α) of some Moran fractals, Monatsh. Math. 144 (2005), 141–155.10.1007/s00605-004-0254-3
Search in Google Scholar Back to article

Certain Singular Distributions and Fractals

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