Simplicial Approach to Fractal Structures

Contemporary Mathematics, 2013

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Journal of Mathematical Analysis and Applications, 2008

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V 0 = {p 1 , p 2 , p 3 } be the set of vertices of SG and u i (x) = 1 2 (x + p i ) the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations u w = u w 1 u w 2 · · · u w n for any sequence w = (w 1 , w 2 , . . . , w n ) ∈ {1, 2, 3} n . The union of the images of V 0 under these iterations is the set of nth stage vertices V n of SG. Let F : V n → R be any function. Given any numbers α w (w ∈ {1, 2, 3} n ) with 0 < |α w | < 1, there exists a unique continuous extension f : SG → R of F , such that

Filomat, 2020

The aim of this paper is to introduce a topological model of fractals. Self similar fractals will be approached as inverse limit of finite one dimensional topological spaces with alpha continuous bonding functions. The second approach is to investigate topological graphs in terms nano topological spaces for Lellis Thivagar. From these approximations, the dynamics of Julia sets as a special type of self similar fractals will be studied and some physical properties of fractals through their nano topological graphs will be applied.

Proceedings of symposia in pure mathematics, 2004

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In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Fractal and Fractional

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D&gt;d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

Journal of Physics A-mathematical and General, 2005

We study Hamiltonian walks (HWs) on Sierpinski and $n$--simplex fractals. Via numerical analysis of exact recursion relations for the number of HWs we calculate the connectivity constant $\omega$ and find the asymptotic behaviour of the number of HWs. Depending on whether or not the polymer collapse transition is possible on a studied lattice, different scaling relations for the number of HWs are obtained. These relations are in general different from the well-known form characteristic of homogeneous lattices which has thus far been assumed to hold for fractal lattices too.

arXiv (Cornell University), 2023

In this article, we show that α-fractal functions defined on Sierpiński gasket (denoted by) depend continuously on the parameters involved in the construction. In the latter part of this article, the continuous dependence of parameters on α-fractal functions defined on is shown graphically.

Fractals in Engineering, 2005

We consider an iterated function system (IFS) of one-to-one contractive maps on a compact metric space. We define the top of an IFS; define an associated symbolic dynamical system; present and explain a fast algorithm for computing the top; describe an example in one dimension with a rich history going back to work of A.Rényi [Representations for Real Numbers and Their Ergodic Properties, Acta Math. Acad. Sci. Hung.,8 (1957), pp. 477-493]; and we show how tops may be used to help to model and render synthetic pictures in applications in computer graphics.