More about Cantor like sets in hyperbolic numbers. (English) Zbl 1375.28019
Summary: In this paper, we discuss the construction of new Cantor like sets in the hyperbolic plane. Also, we study the arithmetic sum of two of these Cantor like sets, as well as of those previously introduced in the literature. An hyperbolization, in the sense of Gromov, of the commutative ring of hyperbolic numbers is also given. Finally, we present the construction of a Cantor-type set as hyperbolic boundary.
MSC:
| 28A80 | Fractals |
Keywords:
hyperbolic numbers; Cantor set; fractal carpets; Gromov hyperbolic space; hyperbolic boundaryReferences:
| [1] | J. Cockle, III, On a new imaginary in algebra, Philos. Mag. Ser. 334(226) (1849) 37-47. · JFM 04.0487.02 |
| [2] | Jancewicz, B., The extended Grassmann algebra of \(\mathbb{R}^3\), in Clifford (Geometric) Algebras with Applications in Physics (ed.) Baylis, W. E., (Birkhäuser, Boston, 1996), pp. 389-421. · Zbl 0861.15044 |
| [3] | Alpay, D., Luna-Elizarrarás, M. E., Shapiro, M. and Struppa, D. C., Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis (Springer Science, Business Media, 2014). · Zbl 1319.46001 |
| [4] | Luna-Elizarrarás, M. E., Shapiro, M., Struppa, D. C. and Vajiac, A., Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers (Birkhäuser/Springer, Cham, 2015). · Zbl 1345.30002 |
| [5] | Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E. and Zampatti, P., The mathematics of Minkowski space-time (Birkhauser, Basel, 2008). · Zbl 1151.53001 |
| [6] | Gargoubi, H. and Kossentini, S., f-algebra structure on hyperbolic numbers, Adv. Appl. Clifford Algebras26(4) (2016) 1211-1233. · Zbl 1403.06028 |
| [7] | Motter, A. E. and Rosa, M. A. F., Hyperbolic calculus, Adv. Appl. Clifford Algebras8(1) (1998) 109-128. · Zbl 0922.30035 |
| [8] | Cantor, G., De la puissance des ensembles parfaits de points, Acta Math.4(1) (1884) 349-362. · JFM 16.0460.01 |
| [9] | Mandelbrot, B., The Fractal Geometry of Nature, 18th printing (Freeman, New York, 1999). |
| [10] | Taylor, T. D., Hudson, C. and Anderson, A., Examples of using binary Cantor sets to study the connectivity of Sierpinski relatives, Fractals20(1) (2012) 61-75. · Zbl 1242.28017 |
| [11] | Bunde, A. and Havlin, S., Fractals in Science (Springer, New York, 1994). · Zbl 0796.00013 |
| [12] | Kennedy, J. A. and Yorke, J. A., Bizarre topology is natural in dynamical systems, Bull. Am. Math. Soc. (N.S.)32(3) (1995) 309316. · Zbl 0868.58055 |
| [13] | Balankin, A. S., Reyes, J. Bory, Luna-Elizarrarás, M. E. and Shapiro, M., Cantor-type sets in hyperbolic numbers, Fractals24(4) (2016) 1650051. · Zbl 1357.28008 |
| [14] | H. Saini and R. Kumar, Some fundamental theorems of functional analysis with bicomplex and hyperbolic scalars, arXiv:1510.01538v2 [math.FA]. · Zbl 1451.30096 |
| [15] | Sobczyk, G., The hyperbolic number plane, College Math. J.26(4) (1995) 268-280. |
| [16] | Sander, L. M., Fractal growth, Sci. Am.256(1) (1987) 94-100. |
| [17] | D. H. Werner and S. Gargiel, An overview of fractal antenna engineering research, IEEE Antennas Propag. Mag.45(1) (2013) 38-57. |
| [18] | Barnsley, M., Fractals Everywhere (Academic Press, Inc.Boston, MA, 1988). · Zbl 0691.58001 |
| [19] | E. W. Weisstein, Haferman Carpet, From MathWorldA Wolfram Web Resource, http://mathworld.wolfram.com/HafermanCarpet.html. |
| [20] | Palis, J. and Takens, F., Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors, , Vol. 35 (Cambridge University Press, Cambridge, 1993). · Zbl 0790.58014 |
| [21] | Gelbaum, B. R. and Olmsted, J. M. H., Counterexamples in Analysis (Dover Publications, New York, 2003). · Zbl 1085.26002 |
| [22] | Burago, D., Burago, Y. and Ivanov, S., A Course in Metric Geometry, , Vol. 33 (American Mathematical Society, Providence, RI, 2001). · Zbl 0981.51016 |
| [23] | Roe, J., Lectures on Coarse Geometry, , Vol. 31 (American Mathematical Society, Providence, RI, 2003). · Zbl 1042.53027 |
| [24] | Ibragimov, Z. and Simanyi, J., Hyperbolic construction of Cantor sets, Involve6(3) (2013) 333-343. · Zbl 1408.30026 |
| [25] | Yingqing, X. and Junping, G., Uniform Cantor sets as hyperbolic boundaries, Filomat28(8) (2014) 1737-1745. · Zbl 1474.30169 |
| [26] | Luna-Elizarrarás, M. E., Pérez-Regalado, C. O. and Shapiro, M., On the Laurent series for bicomplex holomorphic functions, Complex Var. Elliptic Equ. (2017), doi :10.1080/17476933.2016.1250404. · Zbl 1375.30077 |
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