Sierpiński gasket as a Martin boundary. II. The intrinsic metric

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.

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GEODESICS OF THE SIERPINSKI GASKET

Fractals ◽

10.1142/s0218348x1850024x ◽

2018 ◽

Vol 26 (03) ◽

pp. 1850024 ◽

Cited By ~ 5

Author(s):

MUSTAFA SALTAN ◽

YUNUS ÖZDEMİR ◽

BÜNYAMİN DEMİR

Keyword(s):

Sierpinski Gasket ◽

Intrinsic Metric ◽

Sierpiński Gasket

In this paper, we examine the number of geodesics between two points of the Sierpinski Gasket ([Formula: see text]) via code representations of the points and as a main result we show that the maximum number of geodesics between different two points with respect to the intrinsic metric on [Formula: see text] is five.

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An explicit formula of the intrinsic metric on the Sierpinski gasket via code representation

TURKISH JOURNAL OF MATHEMATICS ◽

10.3906/mat-1702-55 ◽

2018 ◽

Vol 42 (2) ◽

Cited By ~ 6

Author(s):

MUSTAFA SALTAN ◽

YUNUS ÖZDEMİR ◽

BÜNYAMİN DEMİR

Keyword(s):

Explicit Formula ◽

Sierpinski Gasket ◽

Intrinsic Metric ◽

Sierpiński Gasket

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Martin Metrics on the Sierpiński Gasket

Stochastics and Dynamics ◽

10.1142/s021949370300067x ◽

2003 ◽

Vol 03 (02) ◽

pp. 267-277

Author(s):

Atsushi Imai ◽

Yasuhiro Kawasaki ◽

Hiroshi Sato

Keyword(s):

Dynamical System ◽

Markov Chain ◽

Topological Space ◽

Sierpinski Gasket ◽

Martin Boundary ◽

Sierpiński Gasket ◽

Hausdorff Dimensions ◽

Lipschitz Equivalence ◽

Euclidean Metric ◽

Affine Mappings

It is known that the (N–1)-dimensional Sierpiński gasket [Formula: see text] is defined as an attractor of a dynamical system induced by affine mappings fi (i = 1, 2, …, N). Identifying fi with i, Denker and Sato [1] represented [Formula: see text] as the Martin boundary of a Markov chain on the word space generated by N letters. The Martin metrics determine a unique topological space on [Formula: see text] but are not always Lipschitz equivalent. In this paper, we characterize the Lipschitz equivalence of Martin metrics in terms of their coefficients, compare them with the Euclidean metric and compute their Hausdorff dimensions.

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