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.

scholarly journals SPECTRAL SELF-AFFINE MEASURES IN $\mathbb{R}^N$

2007 ◽  
Vol 50 (1) ◽  
pp. 197-215 ◽  
Author(s):  
Jian-Lin Li
Keyword(s):  
Large Class ◽  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Iterated Function Systems ◽  
Sierpiński Gasket ◽  
Spectral Pair ◽  
Iterated Function ◽  
Spectral Pairs ◽  
The Given ◽  
Check Condition

AbstractThe aim of this paper is to investigate and study the possible spectral pair $(\mu_{M,D},\varLambda(M,S))$ associated with the iterated function systems $\{\phi_{d}(x)= M^{-1}(x+d)\}_{d\in D}$ and $\{\psi_{s}(x)=M^{\ast}x+s\}_{s\in S}$ in $\mathbb{R}^n$. For a large class of self-affine measures $\mu_{M,D}$, we obtain an easy check condition for $\varLambda(M,S)$ not to be a spectrum, and answer a question of whether we have such a spectral pair $(\mu_{M,D},\varLambda(M,S))$ in the Eiffel Tower or three-dimensional Sierpinski gasket. Further generalization of the given condition as well as some elementary properties of compatible pairs and spectral pairs are discussed. Finally, we give several interesting examples to illustrate the spectral pair conditions considered here.

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 Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

scholarly journals Ruelle operator theorem for non-expansive systems

2009 ◽  
Vol 30 (2) ◽  
pp. 469-487 ◽  
Author(s):  
YUNPING JIANG ◽  
YUAN-LING YE
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Iterated Function Systems ◽  
Ruelle Operator ◽  
Iterated Function

AbstractThe Ruelle operator theorem has been studied extensively both in dynamical systems and iterated function systems. In this paper we study the Ruelle operator theorem for non-expansive systems. Our theorems give some sufficient conditions for the Ruelle operator theorem to be held for a non-expansive system.

scholarly journals MULTIRESOLUTION CONTROL OF CURVES AND SURFACES WITH A SELF-SIMILAR MODEL

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 271-286 ◽  
Author(s):  
HOUSSAM HNAIDI ◽  
ERIC GUÉRIN ◽  
SAMIR AKKOUCHE
Keyword(s):  
Wavelet Transform ◽  
Control Point ◽  
Iterated Function Systems ◽  
Control Points ◽  
Similar Model ◽  
Surface Theory ◽  
Curves And Surfaces ◽  
Iterated Function ◽  
Resolution Level ◽  
Self Similar

This paper presents two self-similar models that allow the control of curves and surfaces. The first model is based on IFS (Iterated Function Systems) theory and the second on subdivision curve and surface theory. Both of these methods employ the detail concept as in the wavelet transform, and allow the multiresolution control of objects with control points at any resolution level.In the first model, the detail is inserted independently of control points, requiring it to be rotated when applying deformations. In contrast, the second method describes details relative to control points, allowing free control point deformations.Modeling examples of curves and surfaces are presented, showing manipulation facilities of the models.

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
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