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

Melis Güneri,Mustafa Saltan

doi:10.3390/fractalfract3010013

Melis Güneri, Mustafa Saltan

Open Access

https://doi.org/10.3390/fractalfract3010013

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Abstract

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.

Highlights

Fractal geometry is one of the most remarkable developments in mathematics in recent years
A geometric shape is self-similar if there is a point such that every neighborhood of the point contains a copy of the entire shape and, if it is self-similar at every point, it is called strictly self-similar
In [12], the intrinsic metric on the code set of the Sierpinski gasket, which is one of the instructive examples of strictly self-similar sets, is formulated by the code representations of its points. Due to this metric formula, important geometrical and topological properties of the Sierpinski gasket are expressed by the code sets, the number of geodesics are determined and the code representations of points are classified according to the number of geodesics

Summary

Introduction

Fractal geometry is one of the most remarkable developments in mathematics in recent years. In [12], the intrinsic metric on the code set of the Sierpinski gasket, which is one of the instructive examples of strictly self-similar sets, is formulated by the code representations of its points Due to this metric formula, important geometrical and topological properties of the Sierpinski gasket are expressed by the code sets, the number of geodesics are determined and the code representations of points are classified according to the number of geodesics (for details, see [13,14,15]). As seen in these studies, defining the intrinsic metrics by using the code representations of the points on fractals provides some facilities for different works. (for details, see [12,15])

The Code Representations of Points on the Sierpinski Propellers

The Construction of the Intrinsic Metric on the Code Set of SP

The Intrinsic Metric Formula on Two Adjacent Right Sierpinski Gaskets

Conclusions