EXAMPLES OF USING BINARY CANTOR SETS TO STUDY THE CONNECTIVITY OF SIERPIŃSKI RELATIVES

Fractals ◽  
2012 ◽  
Vol 20 (01) ◽  
pp. 61-75 ◽  
Author(s):  
T. D. TAYLOR ◽  
C. HUDSON ◽  
A. ANDERSON
Keyword(s):  
Fractal Dimension ◽  
Sierpinski Gasket ◽  
Cantor Set ◽  
Cantor Sets ◽  
Sierpiński Gasket ◽  
Multiply Connected ◽  
Simply Connected ◽  
Totally Disconnected

The Sierpiński relatives form a class of fractals that all have the same fractal dimension, but different topologies. This class includes the well-known Sierpiński gasket. Some relatives are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. This paper presents examples of relatives for which binary Cantor sets are relevant for the connectivity. These Cantor sets are variations of the usual middle thirds Cantor set, and their binary descriptions greatly aid in the determination of the connectivity of the corresponding relatives.

CONNECTIVITY PROPERTIES OF SIERPIŃSKI RELATIVES

Fractals ◽  
2011 ◽  
Vol 19 (04) ◽  
pp. 481-506 ◽  
Author(s):  
T. D. TAYLOR
Keyword(s):  
Fractal Dimension ◽  
Multiply Connected ◽  
Simply Connected ◽  
Totally Disconnected

This paper presents a study of the connectivity of the class of fractals known as the Sierpiński relatives. These fractals all have the same fractal dimension, but different topologies. Some are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. Conditions for these four cases are presented. Constructions of paths, including non-contractible closed paths in the case of multiply-connected relatives, are presented. Examples of specific relatives are provided to illustrate the four cases.

scholarly journals Exotic Nonleaves with Infinitely Many Ends

2021 ◽  
Author(s):  
Carlos Meniño Cotón ◽  
Paul A Schweitzer
Keyword(s):  
Compact Manifold ◽  
Cantor Set ◽  
Remarkable Case ◽  
Realization Problem ◽  
Codimension One ◽  
Simply Connected ◽  
New Criterion ◽  
Totally Disconnected

Abstract We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $\Lambda $ admits a continuum of smoothings, which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ punctured along a tame Cantor set. This is the lowest reasonable regularity for this realization problem. These results come from a new criterion for nonleaves in $C^{1,0}$ regularity. We also include a new criterion for nonleaves in the $C^2$-category. Some of our smooth nonleaves are “exotic”, that is, homeomorphic but not diffeomorphic to leaves of codimension one foliations on a compact manifold, being the 1st examples in this class.

scholarly journals Computability of the packing measure of totally disconnected self-similar sets

2015 ◽  
Vol 36 (5) ◽  
pp. 1534-1556 ◽  
Author(s):  
MARTA LLORENTE ◽  
MANUEL MORÁN
Keyword(s):  
Sierpinski Gasket ◽  
Separation Condition ◽  
Packing Measure ◽  
Sierpiński Gasket ◽  
Additional Information ◽  
Strong Separation ◽  
Self Similar ◽  
Contraction Factor ◽  
Strong Separation Condition ◽  
Totally Disconnected

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called Strong Separation Condition (SSC) and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contraction factor in the interval $(0,\frac{1}{3}]$.

Optimal Density Determination of Bouguer Anomaly Using Fractal Analysis (Case of study: Charak area,IRAN)

2005 ◽  
Vol 1 (1) ◽  
pp. 21-24
Author(s):  
Hamid Reza Samadi
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Host Rock ◽  
Bouguer Anomaly ◽  
Research Area ◽  
Density Difference ◽  
Optimal Density

In exploration geophysics the main and initial aim is to determine density of under-research goals which have certain density difference with the host rock. Therefore, we state a method in this paper to determine the density of bouguer plate, the so-called variogram method based on fractal geometry. This method is based on minimizing surface roughness of bouguer anomaly. The fractal dimension of surface has been used as surface roughness of bouguer anomaly. Using this method, the optimal density of Charak area insouth of Hormozgan province can be determined which is 2/7 g/cfor the under-research area. This determined density has been used to correct and investigate its results about the isostasy of the studied area and results well-coincided with the geology of the area and dug exploratory holes in the text area

scholarly journals Dimer Coverings on the Sierpinski Gasket

2008 ◽  
Vol 131 (4) ◽  
pp. 631-650 ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket

scholarly journals Metric approximations of spectral triples on the Sierpiński gasket and other fractal curves

2021 ◽  
Vol 385 ◽  
pp. 107771
Author(s):  
Therese-Marie Landry ◽  
Michel L. Lapidus ◽  
Frédéric Latrémolière
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Spectral Triples
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