GEODESICS OF HIGHER-DIMENSIONAL SIERPINSKI GASKET

It is of great interest to analyze geodesics in fractals. We investigate the structure of geodesics in [Formula: see text]-dimensional Sierpinski gasket [Formula: see text] for [Formula: see text], and prove that there are at most eight geodesics between any pair of points in [Formula: see text]. Moreover, we obtain that there exists a unique geodesic for almost every pair of points in [Formula: see text].

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Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets

Canadian Journal of Mathematics ◽

10.4153/cjm-2009-054-5 ◽

2010 ◽

Vol 61 (5) ◽

pp. 1151-1181 ◽

Cited By ~ 1

Author(s):

Huo-Jun Ruan ◽

Robert S. Strichartz

Keyword(s):

Fourier Series ◽

Periodic Function ◽

Sierpinski Gasket ◽

Series Expansions ◽

Periodic Functions ◽

Sierpiński Gasket ◽

Higher Dimensional ◽

Sierpinski Gaskets

Abstract.We construct covering maps from infinite blowups of the$n$-dimensional Sierpinski gasket$S{{G}_{n}}$to certain compact fractafolds based on$S{{G}_{n}}$. These maps are fractal analogs of the usual covering maps fromthe line to the circle. The construction extends work of the second author in the case$n=2$, but a differentmethod of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

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MEAN VALUE PROPERTY OF HARMONIC FUNCTION ON THE HIGHER-DIMENSIONAL SIERPINSKI GASKET

Fractals ◽

10.1142/s0218348x20500772 ◽

2020 ◽

Vol 28 (05) ◽

pp. 2050077

Author(s):

YIPENG WU ◽

ZHILONG CHEN ◽

XIA ZHANG ◽

XUDONG ZHAO

Keyword(s):

Harmonic Function ◽

Harmonic Functions ◽

Sierpinski Gasket ◽

Mean Value ◽

Sierpiński Gasket ◽

Mean Value Property ◽

Average Value ◽

Higher Dimensional ◽

The Mean

Harmonic functions possess the mean value property, that is, the value of the function at any point is equal to the average value of the function in a domain that contain this point. It is a very attractive problem to look for analogous results in the fractal context. In this paper, we establish a similar results of the mean value property for the harmonic functions on the higher-dimensional Sierpinski gasket.

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HARMONIC GRADIENTS ON HIGHER-DIMENSIONAL SIERPIŃSKI GASKETS

Fractals ◽

10.1142/s0218348x2050108x ◽

2020 ◽

Vol 28 (06) ◽

pp. 2050108

Author(s):

LUKE BROWN ◽

GIOVANNI FERRER ◽

GAMAL MOGRABY ◽

LUKE G. ROGERS ◽

KARUNA SANGAM

Keyword(s):

Dirichlet Form ◽

Sierpinski Gasket ◽

Sharp Contrast ◽

Standard Measure ◽

Sierpiński Gasket ◽

Higher Dimensional ◽

Sierpinski Gaskets ◽

Funct Anal

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants [Formula: see text], [Formula: see text], proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal. 174(1) (2000) 128–154]. When [Formula: see text] is equipped with the standard Dirichlet form and measure [Formula: see text] we show there is a full [Formula: see text]-measure set on which continuity of the Laplacian implies existence of the gradient [Formula: see text], and that this set is not all of [Formula: see text]. We also show there is a class of non-uniform measures on the usual Sierpiński Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere in sharp contrast to the case with the standard measure.

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