Abstract
In the first part of this work we study the set of bicomplex numbers from the point of view of a hyperbolic module. We make use of the partial order defined on the set of hyperbolic numbers. We recall some properties of the hyperbolic geometrical objects that were defined in previous papers. With the help of these notions some fractal-type sets in the four dimensional space are constructed.
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Alpay, D., Luna, M.E., Shapiro, M., Struppa, D.C.: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis. Series SpringerBriefs in Mathematics. Springer, Cham (2014)
Balankin, A.S.: Physics in space-time with scale-dependent metrics. Phys. Lett. A 377, 1606–1610 (2013)
Balankin, A.S.: Effective degrees of freedom of a random walk on a fractal. Phys. Rev. E 92, 062146 (2015)
Balankin, A.S., Bory-Reyes, J., Luna-Elizarrarás, M.E., Shapiro, M.: Cantor-type sets in hyperbolic numbers. Fractals 24(04), 1650051 (2016)
Eardley, D.M.: Self-similar spacetimes: geometry and dynamics. Commun. Math. Phys. 37, 287–309 (1974)
He, J.H.: Hilbert cube model for fractal spacetime. Chaos Solitons Fractals 42, 2754–2759 (2009)
Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
Kumar, R., Saini, H.: Topological bicomplex modules. Adv. Appl. Clifford Algebras 26, 1249–1270 (2016)
Kocić, Ljubiša M., Matejić, Marjan M.: Contractive Affine transformations of complex plane and applications. Facta Universitatis (NIŠ) Ser. Math. Inform. 21, 65–75 (2006)
Lichtenberg, A.J., Liebermann, M.A.: Regular and Chaotic Dynamics, 2nd edn. Springer, New York (1992)
Luna-Elizarrarás, M.E., Pérez-Regalado, C.O., Shapiro, M.: On linear functionals and Hahn–Banach theorems for hyperbolic and bicomplex modules. Adv. Appl. Clifford Algebras 24(4), 1105–1129 (2014)
Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex numbers and their elementary functions. Cubo A Math. J. 14(2), 61–80 (2012)
Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers. Frontiers in Mathematics. Springer, Birkhäuser (2015)
Mandelbrot, B.: The Fractal Geometry of Nature. Freeman, New York (1999)
Nikiel, S.: Iterated Function Systems for Real-Time Image Synthesis. Springer, London (2007)
Richter, M., Lange, S., Backer, A., Ketzmerick, R.: Visualization and comparison of classical structures and quantum states of four-dimensional maps. Phys. Rev. E 89, 022902 (2014)
Rochon, D.: A generalized Mandelbrot set for bicomplex numbers. Fractals 8(04), 355–368 (2000)
Shapiro, M., Struppa, D.C., Vajiac, A., Vajiac, M.B.: Hyperbolic numbers and their functions. Anal. Univ. Oradea Fasc. Math. XIX(1), 265–283 (2012)
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Luna-Elizarrarás, M.E., Shapiro, M. & Balankin, A. Fractal-type sets in the four-dimensional space using bicomplex and hyperbolic numbers. Anal.Math.Phys. 10, 13 (2020). https://doi.org/10.1007/s13324-020-00356-5
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DOI: https://doi.org/10.1007/s13324-020-00356-5