Abstract
Checking the similarity of some Julia sets and Mandelbrot sub areas in Misiurewicz points by calculating their fractal dimension is the main purpose of this paper. MATLAB programs are used to generate the Julia sets images that match the Misiurewicz points; these images are the entry to the FracLac software. Using this software we were able to find different measurements that characterize those fractals in textures and other features. We are actually focusing on fractal dimension and the error calculated by the software. When performing the given equation of regression or the logarithmic slope of an image, a Box Counting method is applied to the entire image, and the chosen features (grid design, scaling method, number of grid positions…) are available in a FracLac Program, then we attempt to prepare the appropriate settings to get the best performance of the software. Finally, a comparison is done for each image corresponding to the area (boundary) where the Misiurewicz point is located.
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References
Les fractales: Art, Nature et Modélisation, Tangente Hors-série no. 8, Editions Pôles (2004)
Milnor, J.: Self-similarity and hairiness in the Mandelbrot set. Comput. Geom. Topol. 114, 211–257 (1989)
Fraser, J.: An Introduction to Julia Sets (2009)
Lajoie, J.: La géométrie fractale. Diss. Université du Québec à Trois-Rivières (2006)
Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, England (2004)
Peitgen, H.-O., Jürgens, H., Saupe, D.: Chaos and Fractals: New Frontiers of Science, 2nd edn. Springer, Science & Business Media, Verlag, New York (2006)
Lei, T.: Voisinages connexes des points de Misiurewicz. Annales de l’institut Fourier 42(4), 707–735 (1992)
Douady, A., Hubbard, J.H.: Itération des pôlynomes quadratiques complexes. CRAS Paris 294, 123–126 (1982)
Douady, A., Hubbard, J.H.: Etude dynamique des polynômes complexes. Part II. Publication mathématique d’Orsay, 85–02 (1985)
Fraclac Homepage: https://imagej.nih.gov/ij/plugins/fraclac/fraclac.html. Last accessed 23 July 2018
Karperien, A.: FracLac for ImageJ. http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm (1999–2013)
Mandelbrot, B.B., Pignoni, R.: The Fractal Geometry of Nature. Revised and enlarged ed. WH freeman and Co, New York (1983)
Lei, T.: Similarity between the Mandelbrot set and Julia sets. Commun. Math. Phys. 134(3), 587–617 (1990)
Shishikura, M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. 147(2), 225–267 (1998)
Bézivin, J.-P.: on the Julia and Fatou sets of ultrametric entire functions. Annals de l’institut Fourier 51(6), 1635–1661 (2001)
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Boussoufi, O., Uahabi, K.L., Atounti, M. (2019). Using Fractal Dimension to Check Similarity Between Mandelbrot and Julia Sets in Misiurewicz Points. In: Ezziyyani, M. (eds) Advanced Intelligent Systems for Sustainable Development (AI2SD’2018). AI2SD 2018. Advances in Intelligent Systems and Computing, vol 915. Springer, Cham. https://doi.org/10.1007/978-3-030-11928-7_50
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DOI: https://doi.org/10.1007/978-3-030-11928-7_50
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