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Journal articles on the topic 'Cantor sets'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 January 2023

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1

Evans, Michael J., Paul D. Humke, and Karen Saxe. "Symmetric porosity of symmetric Cantor sets." Czechoslovak Mathematical Journal 44, no. 2 (1994): 251–64. http://dx.doi.org/10.21136/cmj.1994.128468.

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2

Geschke, Stefan, Jan Grebík, and Benjamin Miller. "Scrambled Cantor sets." Proceedings of the American Mathematical Society 149, no. 10 (July 20, 2021): 4461–68. http://dx.doi.org/10.1090/proc/15532.

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3

Cheraghi, Davoud, and Mohammad Pedramfar. "Hairy Cantor sets." Advances in Mathematics 398 (March 2022): 108168. http://dx.doi.org/10.1016/j.aim.2021.108168.

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4

Berger, Pierre, and Carlos Gustavo Moreira. "Nested Cantor sets." Mathematische Zeitschrift 283, no. 1-2 (January 21, 2016): 419–35. http://dx.doi.org/10.1007/s00209-015-1605-6.

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5

栾, 佳璇. "Research on the Properties of Cantor Sets and Cantor Functions." Advances in Applied Mathematics 10, no. 04 (2021): 1222–28. http://dx.doi.org/10.12677/aam.2021.104132.

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6

Fletcher, Alastair, and Jang-Mei Wu. "Julia sets and wild Cantor sets." Geometriae Dedicata 174, no. 1 (September 17, 2014): 169–76. http://dx.doi.org/10.1007/s10711-014-0010-3.

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7

Grines, V. Z., and E. V. Zhuzhoma. "Cantor Type Basic Sets of Surface $A$-endomorphisms." Nelineinaya Dinamika 17, no. 3 (2021): 335–45. http://dx.doi.org/10.20537/nd210307.

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The paper is devoted to an investigation of the genus of an orientable closed surface $M^{2}$ which admits $A$-endomorphisms whose nonwandering set contains a one-dimensional strictly invariant contracting repeller $\Lambda_{r}$ with a uniquely defined unstable bundle and with an admissible boundary of finite type. First, we prove that, if $M^{2}$ is a torus or a sphere, then $M^{2}$ admits such an endomorphism. We also show that, if $\Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^{2}\to M^{2}$ of a closed orientable surface $M^{2}$ and $f$ is not a diffeomorphism, then $\Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^{2}\to M^{2}$ is an $A$-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type $\Omega_{r}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_{r}$ is regular, then $M^{2}$ is a two-dimensional torus $\mathbb{T}^{2}$ or a two-dimensional sphere $\mathbb{S}^{2}$.

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8

Wright, David. "Bing-Whitehead Cantor sets." Fundamenta Mathematicae 132, no. 2 (1989): 105–16. http://dx.doi.org/10.4064/fm-132-2-105-116.

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9

PERES, YUVAL, and PABLO SHMERKIN. "Resonance between Cantor sets." Ergodic Theory and Dynamical Systems 29, no. 1 (February 2009): 201–21. http://dx.doi.org/10.1017/s0143385708000369.

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AbstractLet Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

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10

CABRELLI, CARLOS A., KATHRYN E. HARE, and URSULA M. MOLTER. "Sums of Cantor sets." Ergodic Theory and Dynamical Systems 17, no. 6 (December 1997): 1299–313. http://dx.doi.org/10.1017/s0143385797097678.

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We find conditions on the ratios of dissection of a Cantor set so that the group it generates under addition has positive Lebesgue measure. In particular, we answer affirmatively a special case of a conjecture posed by J. Palis.

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11

BAMÓN, RODRIGO, CARLOS G. MOREIRA, SERGIO PLAZA, and JAIME VERA. "Differentiable structures of central Cantor sets." Ergodic Theory and Dynamical Systems 17, no. 5 (October 1997): 1027–42. http://dx.doi.org/10.1017/s014338579708629x.

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Central Cantor sets form a class of symmetric Cantor sets of the real line. Here we give a complete characterization of the $C^{k + \alpha}$ regularity of these Cantor sets. We also give a classification of central Cantor sets up to global and local diffeomorphisms. Examples of central Cantor sets with special dynamical and measure-theoretical properties are also provided. Finally, we calculate the fractal dimensions of an arbitrary central Cantor set.

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12

Hare, Kathryn E. "Maximal Operators and Cantor Sets." Canadian Mathematical Bulletin 43, no. 3 (September 1, 2000): 330–42. http://dx.doi.org/10.4153/cmb-2000-040-5.

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13

Nymann, J. "Linear combinations of Cantor sets." Colloquium Mathematicum 68, no. 2 (1995): 259–64. http://dx.doi.org/10.4064/cm-68-2-259-264.

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14

Cui, Keqin, Wenjia Ma, and Kan Jiang. "Geometric progressions meet Cantor sets." Chaos, Solitons & Fractals 163 (October 2022): 112567. http://dx.doi.org/10.1016/j.chaos.2022.112567.

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15

Dubuc. "A NOTE ON CANTOR SETS." Real Analysis Exchange 23, no. 2 (1997): 767. http://dx.doi.org/10.2307/44153998.

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16

Dettmann, C. P., N. E. Frankel, and T. Taucher. "Structure factor of Cantor sets." Physical Review E 49, no. 4 (April 1, 1994): 3171–78. http://dx.doi.org/10.1103/physreve.49.3171.

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17

Takahashi, Yuki. "Products of two Cantor sets." Nonlinearity 30, no. 5 (April 10, 2017): 2114–37. http://dx.doi.org/10.1088/1361-6544/aa6761.

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18

Khalili Golmankhaneh, Ali, Saleh Ashrafi, Dumitru Baleanu, and Arran Fernandez. "Brownian Motion on Cantor Sets." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 3-4 (May 26, 2020): 275–81. http://dx.doi.org/10.1515/ijnsns-2018-0384.

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AbstractIn this paper, we have investigated the Langevin and Brownian equations on fractal time sets using Fα-calculus and shown that the mean square displacement is not varied linearly with time. We have also generalized the classical method of deriving the Fokker–Planck equation in order to obtain the Fokker–Planck equation on fractal time sets.

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19

Orzechowski, Mark E. "Percolation in Random Cantor Sets." Fractals 05, supp01 (April 1997): 101–9. http://dx.doi.org/10.1142/s0218348x9700067x.

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The d-dimensional random Cantor set is a generalization of the classical "middle-thirds" Cantor set. Starting with the unit cube [0, 1]d, at every stage of the construction we divide each cube remaining into Nd equal subcubes, and select each of these at random with probability p. The resulting limit set is a random fractal C. We present some of the main probabilistic properties of C, with an emphasis on the existence of large connected components ("percolation") and (d-1)-dimensional surfaces ("sheet-percolation"). We also look at some closely related models. Rigorous proofs are not attempted, but we give some heuristic explanations and further references. These notes formed the basis for a talk at the NATO Advanced Study Institute on Fractal Image Encoding and Analysis in Trondheim, Norway on July 8–17, 1995.

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20

Garcia, Ignacio, Ursula Molter, and Roberto Scotto. "Dimension functions of Cantor sets." Proceedings of the American Mathematical Society 135, no. 10 (October 1, 2007): 3151–62. http://dx.doi.org/10.1090/s0002-9939-07-09019-3.

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21

Hunt, Brian R., Ittai Kan, and James A. Yorke. "When Cantor sets intersect thickly." Transactions of the American Mathematical Society 339, no. 2 (February 1, 1993): 869–88. http://dx.doi.org/10.1090/s0002-9947-1993-1117219-8.

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22

Garity, Dennis, Dušan Repovš, David Wright, and Matjaž Željko. "Distinguishing Bing-Whitehead Cantor sets." Transactions of the American Mathematical Society 363, no. 02 (February 1, 2011): 1007. http://dx.doi.org/10.1090/s0002-9947-2010-05175-x.

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23

Astels, S. "Thickness measures for Cantor sets." Electronic Research Announcements of the American Mathematical Society 5, no. 15 (July 20, 1999): 108–11. http://dx.doi.org/10.1090/s1079-6762-99-00068-2.

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24

Crovisier, Sylvain, and Michał Rams. "IFS attractors and Cantor sets." Topology and its Applications 153, no. 11 (May 2006): 1849–59. http://dx.doi.org/10.1016/j.topol.2005.06.010.

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25

Hu, Meidan, and Shengyou Wen. "Quasisymmetrically minimal uniform Cantor sets." Topology and its Applications 155, no. 6 (February 2008): 515–21. http://dx.doi.org/10.1016/j.topol.2007.10.006.

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26

Monterie, M. A. "Capacities of certain Cantor sets." Indagationes Mathematicae 8, no. 2 (June 1997): 247–66. http://dx.doi.org/10.1016/s0019-3577(97)89123-9.

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27

Falconer, K. J. "Projections of random Cantor sets." Journal of Theoretical Probability 2, no. 1 (January 1989): 65–70. http://dx.doi.org/10.1007/bf01048269.

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28

Alpan, Gökalp, and Alexander Goncharov. "Two measures on Cantor sets." Journal of Approximation Theory 186 (October 2014): 28–32. http://dx.doi.org/10.1016/j.jat.2014.07.003.

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29

Zeljko, Matjaz. "On linking of Cantor sets." Glasnik Matematicki 41, no. 1 (June 15, 2006): 165–76. http://dx.doi.org/10.3336/gm.41.1.14.

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30

Kraft, Roger. "Intersections of thick Cantor sets." Memoirs of the American Mathematical Society 97, no. 468 (1992): 0. http://dx.doi.org/10.1090/memo/0468.

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31

Arosio, Leandro, John Erik Fornæss, Nikolay Shcherbina, and Erlend F. Wold. "Squeezing functions and Cantor sets." ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA - CLASSE DI SCIENZE 21, no. 2 (December 2020): 1359–69. http://dx.doi.org/10.2422/2036-2145.201807_003.

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32

Ibragimov, Zair, and John Simanyi. "Hyperbolic construction of Cantor sets." Involve, a Journal of Mathematics 6, no. 3 (September 8, 2013): 333–43. http://dx.doi.org/10.2140/involve.2013.6.333.

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33

Krushkal, Vyacheslav. "Sticky Cantor sets in ℝd." Journal of Topology and Analysis 10, no. 02 (June 2018): 477–82. http://dx.doi.org/10.1142/s1793525318500164.

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A subset of [Formula: see text] is called “sticky” if it cannot be isotoped off of itself by a small ambient isotopy. Sticky wild Cantor sets are constructed in [Formula: see text] for each [Formula: see text].

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34

Bateman, Michael, and Nets Hawk Katz. "Kakeya sets in Cantor directions." Mathematical Research Letters 15, no. 1 (2008): 73–81. http://dx.doi.org/10.4310/mrl.2008.v15.n1.a7.

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35

Veerman, J. J. P. "Intersecting self-similar Cantor sets." Boletim da Sociedade Brasileira de Matem�tica 26, no. 2 (September 1995): 167–81. http://dx.doi.org/10.1007/bf01236992.

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36

Schweizer, J., and B. Frank. "Calculus on linear Cantor sets." Archiv der Mathematik 79, no. 1 (July 2002): 46–50. http://dx.doi.org/10.1007/s00013-002-8283-4.

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37

Bugeaud, Yann. "Diophantine approximation and Cantor sets." Mathematische Annalen 341, no. 3 (February 2, 2008): 677–84. http://dx.doi.org/10.1007/s00208-008-0209-4.

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38

Goncharov, Alexander P. "Weakly Equilibrium Cantor-type Sets." Potential Analysis 40, no. 2 (April 24, 2013): 143–61. http://dx.doi.org/10.1007/s11118-013-9344-y.

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39

Dai, Mei-Feng. "Quasisymmetrically Minimal Moran Sets." Canadian Mathematical Bulletin 56, no. 2 (June 1, 2013): 292–305. http://dx.doi.org/10.4153/cmb-2011-164-2.

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AbstractM. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor sets of Hausdorff dimension 1, where at the k-th set one removes from each interval I a certain number nk of open subintervals of length ck|I|, leaving (nk + 1) closed subintervals of equal length. Quasisymmetrically Moran sets of Hausdorff dimension 1 considered in the paper are more general than uniform Cantor sets in that neither the open subintervals nor the closed subintervals are required to be of equal length.

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40

DEVANEY, ROBERT L. "Cantor sets of circles of Sierpiński curve Julia sets." Ergodic Theory and Dynamical Systems 27, no. 5 (October 2007): 1525–39. http://dx.doi.org/10.1017/s0143385707000156.

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AbstractOur goal in this paper is to give an example of a one-parameter family of rational maps for which, in the parameter plane, there is a Cantor set of simple closed curves consisting of parameters for which the corresponding Julia set is a Sierpiński curve. Hence, the Julia sets for each of these parameters are homeomorphic. However, each of the maps in this set is dynamically distinct from (i.e. not topologically conjugate to) any other map in this set (with only finitely many exceptions). We also show that, in the dynamical plane for any map drawn from a large open set in the connectedness locus in this family, there is a Cantor set of invariant simple closed curves on which the map is conjugate to the product of certain subshifts of finite type with the maps $z \mapsto \pm z^n$ on the unit circle.

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41

Mendivil, F., and T. D. Taylor. "Thin Sets with Fat Shadows: Projections of Cantor Sets." American Mathematical Monthly 115, no. 5 (May 2008): 451–56. http://dx.doi.org/10.1080/00029890.2008.11920549.

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42

Li, Wenwen, Wenxia Li, Junjie Miao, and Lifeng Xi. "Assouad dimensions of Moran sets and Cantor-like sets." Frontiers of Mathematics in China 11, no. 3 (April 1, 2016): 705–22. http://dx.doi.org/10.1007/s11464-016-0539-6.

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43

QIU, WEIYUAN, FEI YANG, and YONGCHENG YIN. "Rational maps whose Julia sets are Cantor circles." Ergodic Theory and Dynamical Systems 35, no. 2 (August 19, 2013): 499–529. http://dx.doi.org/10.1017/etds.2013.53.

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AbstractIn this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.

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44

BALANKIN, A. S., J. BORY-REYES, M. E. LUNA-ELIZARRARÁS, and M. SHAPIRO. "CANTOR-TYPE SETS IN HYPERBOLIC NUMBERS." Fractals 24, no. 04 (December 2016): 1650051. http://dx.doi.org/10.1142/s0218348x16500511.

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The construction of the ternary Cantor set is generalized into the context of hyperbolic numbers. The partial order structure of hyperbolic numbers is revealed and the notion of hyperbolic interval is defined. This allows us to define a general framework of the fractal geometry on the hyperbolic plane. Three types of the hyperbolic analogues of the real Cantor set are identified. The complementary nature of the real Cantor dust and the real Sierpinski carpet on the hyperbolic plane are outlined. The relevance of these findings in the context of modern physics are briefly discussed.

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45

Anisca, Razvan, and Monica Ilie. "A Technique of Studying Sums of Central Cantor Sets." Canadian Mathematical Bulletin 44, no. 1 (March 1, 2001): 12–18. http://dx.doi.org/10.4153/cmb-2001-002-8.

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AbstractThis paper is concernedwith the structure of the arithmetic sum of a finite number of central Cantor sets. The technique used to study this consists of a duality between central Cantor sets and sets of subsums of certain infinite series. One consequence is that the sum of a finite number of central Cantor sets is one of the following: a finite union of closed intervals, homeomorphic to the Cantor ternary set or an M-Cantorval.

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46

Johnson, Stewart D. "Absorbing cantor sets and trapping structures." Ergodic Theory and Dynamical Systems 11, no. 4 (December 1991): 731–36. http://dx.doi.org/10.1017/s0143385700006441.

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AbstractIt it shown that a minimal attractor for a continuous, lebesgue non-singular transformation on an interval with no wandering intervals is either a periodic orbit, a finite collection of intervals, a simply attracting cantor set, or an absorbing cantor set.

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47

Zhao, Yang, Dumitru Baleanu, Carlo Cattani, De-Fu Cheng, and Xiao-Jun Yang. "Maxwell’s Equations on Cantor Sets: A Local Fractional Approach." Advances in High Energy Physics 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/686371.

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Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell's equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.

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48

Zhen, Fang-Xiong. "Dimensions of subsets of cantor-type sets." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–8. http://dx.doi.org/10.1155/ijmms/2006/26359.

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49

Rani, Mamta, and Sanjeev Kumar Prasad. "Superior Cantor Sets and Superior Devil Staircases." International Journal of Artificial Life Research 1, no. 1 (January 2010): 78–84. http://dx.doi.org/10.4018/jalr.2010102106.

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Mandelbrot, in 1975, coined the term fractal and included Cantor set as a classical example of fractals. The Cantor set has wide applications in real world problems from strange attractors of nonlinear dynamical systems to the distribution of galaxies in the universe (Schroder, 1990). In this article, we obtain superior Cantor sets and present them graphically by superior devil’s staircases. Further, based on their method of generation, we put them into two categories.

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50

ZENG, YING. "SELF-SIMILAR SUBSETS OF SYMMETRIC CANTOR SETS." Fractals 25, no. 01 (February 2017): 1750003. http://dx.doi.org/10.1142/s0218348x17500037.

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This paper concerns the affine embeddings of general symmetric Cantor sets. Under certain condition, we show that if a self-similar set [Formula: see text] can be affinely embedded into a symmetric Cantor set [Formula: see text], then their contractions are rationally commensurable. Our result supports Conjecture 1.2 in [D. J. Feng, W. Huang and H. Rao, Affine embeddings and intersections of Cantor sets, J. Math. Pures Appl. 102 (2014) 1062–1079].

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