Infinitesimal resistance metrics on Sierpinski gasket type fractals

Analysis ◽  
2008 ◽  
Vol 28 (3) ◽  
Author(s):  
Mihai Cucuringu ◽  
Robert S. Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Periodic Points ◽  
Effective Resistance ◽  
Sierpiński Gasket ◽  
Boundary Points ◽  
Junction Points ◽  
Self Similar ◽  
Zoom In

We prove the existence of an infinitesimal resistance metric on the Sierpinski gasket (SG) at boundary points, junction points and periodic points. This is a renormalized limit of the effective resistance metric as we zoom in on the point, and satisfies a self-similar identity. We obtain similar results on PCF fractals with three boundary points.

scholarly journals On topological conjugacy of some chaotic dynamical systems on the Sierpinski gasket

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2317-2331
Author(s):  
Nisa Aslan ◽  
Mustafa Saltan ◽  
Bünyamin Demir
Keyword(s):  
Dynamical Systems ◽  
Sierpinski Gasket ◽  
Iterated Function Systems ◽  
Time Algorithm ◽  
Periodic Points ◽  
Topological Conjugacy ◽  
Escape Time ◽  
Sierpiński Gasket ◽  
Iterated Function ◽  
Self Similar

The dynamical systems on the classical fractals can naturally be obtained with the help of their iterated function systems. In the recent years, different ways have been developed to define dynamical systems on the self similar sets. In this paper, we give composition functions by using expanding and folding mappings which generate the classical Sierpinski Gasket via the escape time algorithm. These functions also indicate dynamical systems on this fractal. We express the dynamical systems by using the code representations of the points. Then, we investigate whether these dynamical systems are topologically conjugate (equivalent) or not. Finally, we show that the dynamical systems are chaotic in the sense of Devaney and then we also compute and compare the periodic points.

AVERAGE GEODESIC DISTANCE OF NODE-WEIGHTED SIERPINSKI NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950110
Author(s):  
LIFENG XI ◽  
QIANQIAN YE ◽  
JIANGWEN GU
Keyword(s):  
Asymptotic Formula ◽  
Geodesic Distance ◽  
Sierpinski Gasket ◽  
Weight Vector ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar

This paper discusses the asymptotic formula of average distances on node-weighted Sierpinski skeleton networks by using the integral of geodesic distance in terms of self-similar measure on the Sierpinski gasket with respect to the weight vector.

scholarly journals Spectral decimation for families of self-similar symmetric Laplacians on the Sierpiński gasket

2019 ◽  
Vol 7 (1) ◽  
pp. 1-62
Author(s):  
Sizhen Fang ◽  
Dylan King ◽  
Eun Bi Lee ◽  
Robert Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Self Similar

ECCENTRIC DISTANCE SUM OF SIERPIŃSKI GASKET AND SIERPIŃSKI NETWORK

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950016 ◽  
Author(s):  
JIN CHEN ◽  
LONG HE ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Complex Networks ◽  
Sierpinski Gasket ◽  
The Self ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar ◽  
Eccentric Distance

The eccentric distance sum is concerned with complex networks. To obtain the asymptotic formula of eccentric distance sum on growing Sierpiński networks, we study some nonlinear integral in terms of self-similar measure on the Sierpiński gasket and use the self-similarity of distance and measure to obtain the exact value of this integral.

Self–similar Energy Forms on the Sierpinski Gasket with Twists

2007 ◽  
Vol 27 (1) ◽  
pp. 45-60 ◽  
Author(s):  
Mihai Cucuringu ◽  
Robert S. Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Self Similar

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 Polynomials on the Sierpiński gasket with respect to different Laplacians which are symmetric and self-similar

2020 ◽  
Vol 7 (4) ◽  
pp. 387-444
Author(s):  
Christian Loring ◽  
W. Jacob Ogden ◽  
Ely Sandine ◽  
Robert Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Self Similar

scholarly journals Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation

2019 ◽  
Vol 3 (1) ◽  
pp. 13
Author(s):  
Melis Güneri ◽  
Mustafa Saltan
Keyword(s):  
Sierpinski Gasket ◽  
Intrinsic Metric ◽  
Explicit Formulas ◽  
Sierpiński Gasket ◽  
Intrinsic Metrics ◽  
Geometrical Properties ◽  
Self Similar ◽  
Sierpinski Gaskets

In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples.

scholarly journals Explicit Spectral Decimation for a Class of Self-Similar Fractals

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Sergio A. Hernández ◽  
Federico Menéndez-Conde
Keyword(s):  
Laplace Operator ◽  
Sierpinski Gasket ◽  
Higher Dimensions ◽  
Explicit Construction ◽  
Sierpiński Gasket ◽  
Infinite Collection ◽  
Self Similar ◽  
The Laplace Operator

The method of spectral decimation is applied to an infinite collection of self-similar fractals. The sets considered are a generalization of the Sierpinski Gasket to higher dimensions; they belong to the class of nested fractals and are thus very symmetric. An explicit construction is given to obtain formulas for the eigenvalues of the Laplace operator acting on these fractals.

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