Analysis on Fractals

2001 ◽  
Author(s):  
Jun Kigami
Keyword(s):  
Analysis On Fractals

EIGENFORMS ON FRACTALS WITH CONNECTED INTERIOR AND THREE VERTICES

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850082
Author(s):  
MATTEO ELIA ◽  
ROBERTO PEIRONE
Keyword(s):  
Analysis On Fractals

An important problem in analysis on fractals is the existence and the determination of an eigenform on a given finitely ramified fractal. It is known that on every fractal either with three vertices or with connected interior, an eigenform exists for suitable weights on the cells. In this paper, we prove that if the fractal has three vertices and connected interior, the form having all coefficients equal to [Formula: see text] is an eigenform for suitable weights on the cells.

Effective resistances for harmonic structures on p.c.f. self-similar sets

1994 ◽  
Vol 115 (2) ◽  
pp. 291-303 ◽  
Author(s):  
Jun Kigami
Keyword(s):  
Diffusion Processes ◽  
Difference Operator ◽  
Scaling Limit ◽  
Sierpinski Gasket ◽  
Dirichlet Forms ◽  
Sierpiński Gasket ◽  
Know How ◽  
Analysis On Fractals ◽  
Topological Description ◽  
Self Similar

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].

scholarly journals Laplacian, on the Arrowhead Curve

2020 ◽  
Vol 13 (2) ◽  
pp. 19-49
Author(s):  
Claire David
Keyword(s):  
Differential Operator ◽  
Dirichlet Forms ◽  
Specific Measure ◽  
Weak Formulation ◽  
Counting Measure ◽  
Analysis On Fractals ◽  
Starting Point ◽  
Specific Geometry ◽  
The One ◽  
Fractal Domain

In terms of analysis on fractals, the Sierpinski gasket stands out as one of the most studied example. The underlying aim of those studies is to determine a differential operator equivalent to the classic Laplacian. The classically adopted approach is a bidimensional one, through a sequence of so-called prefractals, i.e. a sequence of graphs that converges towards the considered domain. The Laplacian is obtained through a weak formulation, by means of Dirichlet forms, built by induction on the prefractals. It turns out that the gasket is also the image of a Peano curve, the so-called Arrowhead one, obtained by means of similarities from a starting point which is the unit line. This raises a question that appears of interest. Dirichlet forms solely depend on the topology of the domain, and not of its geometry. Which means that, if one aims at building a Laplacian on a fractal domain as the aforementioned curve, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account its specific geometry. Another difference due to the geometry, is encountered may one want to build a specific measure. For memory, the sub-cells of the Kigami and Strichartz approach are triangular and closed: the similarities at stake in the building of the Curve called for semi-closed trapezoids. As far as we know, and until now, such an approach is not a common one, and does not appear in such a context. It intererestingly happens that the measure we choose corresponds, in a sense, to the natural counting measure on the curve. Also, it is in perfect accordance with the one used in the Kigami and Strichartz approach. In doing so, we make the comparison -- and the link -- between three different approaches, that enable one to obtain the Laplacian on the arrowhead curve: the natural method; the Kigami and Strichartz approach, using decimation; the Mosco approach.    

ANALYSIS ON FRACTALS IN FUZZY METRIC SPACES

Fractals ◽  
2011 ◽  
Vol 19 (03) ◽  
pp. 379-386 ◽  
Author(s):  
D. EASWARAMOORTHY ◽  
R. UTHAYAKUMAR
Keyword(s):  
Metric Spaces ◽  
Iterated Function System ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Analysis On Fractals ◽  
Contraction Theorem ◽  
Iterated Function ◽  
Banach Contraction ◽  
Banach Contraction Theorem ◽  
Collage Theorem

In this paper, we investigate the fractals generated by the iterated function system of fuzzy contractions in the fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard fuzzy metric spaces by using the fuzzy Banach contraction theorem. In addition to that, we discuss some results on fuzzy fractals such as Collage Theorem and Falling Leaves Theorem in the standard fuzzy metric spaces with respect to the standard Hausdorff fuzzy metrics.

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