On the topological convergence of multi-rule sequences of sets and fractal patterns
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On the topological convergence of multi-rule sequences of sets and fractal patterns Fabio Caldarola1
· Mario Maiolo2
Published online: 29 October 2020 © The Author(s) 2020
Abstract In many cases occurring in the real world and studied in science and engineering, non-homogeneous fractal forms often emerge with striking characteristics of cyclicity or periodicity. The authors, for example, have repeatedly traced these characteristics in hydrological basins, hydraulic networks, water demand, and various datasets. But, unfortunately, today we do not yet have well-developed and at the same time simple-to-use mathematical models that allow, above all scientists and engineers, to interpret these phenomena. An interesting idea was firstly proposed by Sergeyev in 2007 under the name of “blinking fractals.” In this paper we investigate from a pure geometric point of view the fractal properties, with their computational aspects, of two main examples generated by a system of multiple rules and which are enlightening for the theme. Strengthened by them, we then propose an address for an easy formalization of the concept of blinking fractal and we discuss some possible applications and future work. Keywords Fractal geometry · Hausdorff distance · Topological compactness · Convergence of sets · Möbius function · Mathematical models · Blinking fractals
1 Introduction The word “fractal” was coined by B. Mandelbrot in 1975, but they are known at least from the end of the previous century (Cantor, von Koch, Sierpi´nski, Fatou, Hausdorff, Lévy, etc.). However, it is only in the last few decades that fractals have known a wide and transversal diffusion and the interest of the scientific world towards them has seen an exponential growth. In fact, fractals have been applied in many fields, from the dynamics of chaos to computer science, from signal theory to geology and biology, etc. (see for example Barnsley 1993, 2006; Bertacchini et al. 2018, 2016; Briggs 1992; Falconer 2014; Hastings and SugCommunicated by Yaroslav D. Sergeyev.
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Mario Maiolo [email protected] Fabio Caldarola [email protected]
1
Department of Mathematics and Computer Science, Università della Calabria, Cubo 31/A, 87036 Arcavacata di Rende (CS), Italy
2
Department of Environmental Engineering, Università della Calabria, Cubo 42/B, 87036 Arcavacata di Rende (CS), Italy
ihara 1994; Mandelbrot 1982 and the references therein). Very interesting further links and applications are also those between fractals, space-filling curves and number theory (see, for instance, Caldarola 2018a; Edgar 2008; Falconer 2014; Lapidus and van Frankenhuysen 2000), or fractals and hydrology/hydraulic engineering as we will recall better below. The main characteristic of a fractal, as it is well known, is the property of self-similarity at different scales, and many abstract mathematical models have been created by focusing on this property. In most cases, a fractal is in fact mathematically described by a generating rule or an iterated mechanism, but in the real wor
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