Abstract
This article focuses on the dynamics of the different tridimensional principal slices of the multicomplex Multibrot sets. First, we define an equivalence relation between those slices. Then, we characterize them in order to establish similarities between their behaviors. Finally, we see that any multicomplex tridimensional principal slice is equivalent to a tricomplex slice up to an affine transformation. This implies that, in the context of tridimensional principal slices, Multibrot sets do not need to be generalized beyond the tricomplex space.
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References
Baribeau, L., Ransford, T.: Cross-sections of multibrot sets. J. Anal. 24(1), 95–101 (2016)
Beardon, A.F.: Iteration of Rational Functions: Complex Analytic Dynamical Systems, Volume 132 of Graduate Texts in Mathematics, 1st edn. Springer, New York (1991)
Douady, A., Hubbard, J.H.: Itération des polynômes quadratiques complexes. C.R. Acad. Sci. Paris Série I Math. 294, 123–126 (1982)
Garant-Pelletier, V.: Ensembles de Mandelbrot et de Julia classiques, généralisés aux espaces multicomplexes et théorème de Fatou-Julia généralisé. Master’s thesis, Université du Québec à Trois-Rivières, Canada (2011)
Garant-Pelletier, V., Rochon, D.: On a generalized Fatou–Julia theorem in multicomplex spaces. Fractals 17(3), 241–255 (2009)
Martineau, É.: Bornes de la distance à l’ensemble de Mandelbrot généralisé. Master’s thesis, Université du Québec à Trois-Rivières, Canada (2004)
Martineau, É., Rochon, D.: On a bicomplex distance estimation for the Tetrabrot. Int. J. Bifurc. Chaos 15(9), 3039–3050 (2005)
Meckes, E.S., Meckes, M.W.: Linear Algebra. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge (2018)
Milnor, J.: Dynamics in One Complex Variable : Introductory Lectures, 2nd edn. Springer Vieweg, Berlin (2000)
Parisé, P.-O.: Les ensembles de Mandelbrot tricomplexes généralisés aux polynômes \(\zeta ^p+c\). Master’s thesis, Université du Québec à Trois-Rivières, Canada (2017)
Parisé, P.-O., Ransford, T., Rochon, D.: Tricomplex dynamical systems generated by polynomials of even degree. Chaot. Model. Simul. 1, 37–48 (2017)
Parisé, P.-O., Rochon, D.: A study of dynamics of the tricomplex polynomial \(\eta ^p+ c\). Nonlinear Dyn. 82(1–2), 157–171 (2015)
Parisé, P.-O., Rochon, D.: Tricomplex dynamical systems generated by polynomials of odd degree. Fractals 25(3), 1–11 (2017)
Price, G.B.: An Introduction to Multicomplex Spaces and Functions. M. Dekker, New York (1991)
Rochon, D.: Sur une généralisation des nombres complexes: les tétranombres. Master’s thesis, Université de Montréal, Canada (1997)
Rochon, D.: A generalized Mandelbrot set for bicomplex numbers. Fractals 8(4), 355–368 (2000)
Rochon, D.: A Bloch constant for hyperholomorphic functions. Complex Var. 44, 85–201 (2001)
Rochon, D.: On a generalized Fatou–Julia theorem. Fractals 11(3), 213–219 (2003)
Rochon, D., Shapiro, M.: On algebraic properties of bicomplex and hyperbolic numbers. Anal. Univ. Oradea Fasc. Math. 11(71), 110 (2004)
Sobczyk, G.: The hyperbolic number plane. Coll. Math. J. 26(4), 268–280 (1995)
Struppa, D.C., Vajiac, A., Vajiac, M.B.: Holomorphy in multicomplex spaces. In: Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, pp. 617–634. Springer, Berlin (2012)
Vajiac, A., Vajiac, M.B.: Multicomplex hyperfunctions. Complex Var. Elliptic Equ. 57(7–8), 751–762 (2012)
Wang, X.-Y., Song, W.-J.: The generalized M-J sets for bicomplex numbers. Nonlinear Dyn. 72(1–2), 17–26 (2013)
Acknowledgements
DR is grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC) for financial support. GB would like to thank the FRQNT and the ISM for the awards of graduate research grants. The authors are grateful to Louis Hamel and Étienne Beaulac, from UQTR, for their useful work on the MetatronBrot Explorer in Java.
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Communicated by Wolfgang Sprössig
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Brouillette, G., Rochon, D. Characterization of the Principal 3D Slices Related to the Multicomplex Mandelbrot Set. Adv. Appl. Clifford Algebras 29, 39 (2019). https://doi.org/10.1007/s00006-019-0956-1
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DOI: https://doi.org/10.1007/s00006-019-0956-1