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doi: 10.3934/dcdss.2021004
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On properties of similarity boundary of attractors in product dynamical systems

Nitha Niralda P C , and  Sunil Mathew

Department of Mathematics, National Institute of Technology, Calicut, 673 601, India

* Corresponding author: Nitha Niralda P C

Received  April 2020 Revised  October 2020 Early access January 2021

Fractals in higher dimensional dynamical systems have significant roles in physics and other applied sciences. In this paper, one of the key property of fractals, called self similarity in product systems, is studied using the concept of similarity boundary. The relationship between similarity boundary of an attractor in a product space to one of its projection spaces is discussed. The impact of inverse invariance of similarity boundary on its coordinate iterated function system is analyzed. Fractals satisfying the strong open set condition, restricted to attractors in product spaces, are characterized. The relationship between similarity boundary of attractors in product spaces and their overlapping sets is also obtained. The equivalency of the restricted open set condition (ROSC) and the strong open set condition in product spaces, is proved. Self similarity of an attractor in a product system is characterized using the Hausdorff measure of its similarity boundary. Also, the Hausdorff dimensions of the overlapping set and similarity boundary of attractors for different types of iterated function systems are obtained.

Keywords: Fractal, iterated function system, dynamical system, self similarity, similarity boundary.
Mathematics Subject Classification: 28A80, 31E05.
Citation: Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021004
References:
[1]

R. K. Aswathy and S. Mathew, On different forms of self-similarity, Chaos, Solitons Fractals, 87 (2016), 102-108.  doi: 10.1016/j.chaos.2016.03.021.  Google Scholar

[2]

R. K. Aswathy and S. Mathew, Separation properties of finite products of hyperbolic iterated function systems, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 594-599.  doi: 10.1016/j.cnsns.2018.07.012.  Google Scholar

[3]

J. W. Baish and R. K. Jain, Fractals and cancer, Cancer Research, 60 (2000), 3683-3688.   Google Scholar

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R. Balu and S. Mathew, On $(n, m)$-iterated function systems, Asian-Eur. J. Math., 6 (2013), 1350055, 12pp. doi: 10.1142/S1793557113500551.  Google Scholar

[5]

R. Balu, S. Mathew and N. A. Secelean, Separation properties of $(n,m)$-IFS attractors, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 160-168.  doi: 10.1016/j.cnsns.2017.04.009.  Google Scholar

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[7]

M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A, 399 (1985), 243-275.   Google Scholar

[8]

S. S. Cross, Fractals in pathology, Journal of Pathology, 182 (1997), 1-8.  doi: 10.1002/(SICI)1096-9896(199705)182:1<1::AID-PATH808>3.0.CO;2-B.  Google Scholar

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P. F. Duvall Jr. and L. S. Husch, Attractors of iterated function systems, Proc. Amer. Math. Soc., 116 (1992), 279-284.  doi: 10.1090/S0002-9939-1992-1132850-6.  Google Scholar

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K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

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K. J. Falconer, Generalised dimensions of measures on self-affine sets, Nonlinearity, 12 (1999), 877-891.  doi: 10.1088/0951-7715/12/4/308.  Google Scholar

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K. J. Falconer, Sub self similar sets, Trans. Amer. Math. Soc., 347 (1995), 3121-3129.  doi: 10.1090/S0002-9947-1995-1264809-X.  Google Scholar

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K. J. Falconer, The dimension of self affine fractals II, Math. Proc. Cambridge Philos. Soc., 111 (1992), 169-179.  doi: 10.1017/S0305004100075253.  Google Scholar

[14]

K. J. Falconer, The Hausdorff dimension of self affine fractals, Math. Proc. Cambridge Philos. Soc., 103 (1988), 339-350.  doi: 10.1017/S0305004100064926.  Google Scholar

[15]

K. J. Falconer and J. M. Fraser, The visible part of plane self similar sets, Proc. Amer. Math. Soc., 141 (2013), 269-278.  doi: 10.1090/S0002-9939-2012-11312-7.  Google Scholar

[16]

K. J. Falconer and J. Miao, Random subsets of self affine fractals, Mathematika, 56 (2010), 61-76.  doi: 10.1112/S0025579309000357.  Google Scholar

[17]

K. J. Falconer and J. J. O'Connor, Symmetry and enumeration of self-similar fractals, Bull. Lond. Math. Soc., 39 (2007), 272-282.  doi: 10.1112/blms/bdl031.  Google Scholar

[18]

B. Forte and E. R. Vrscay, Solving the inverse problem for function/image approximation using iterated function systems I: Theoretical basis, Fractals, 2 (1994), 325-334.  doi: 10.1142/S0218348X94000429.  Google Scholar

[19]

B. Forte and E. R. Vrscay, Theory of generalized fractal transforms, Fractal Image Encoding and Analysis, NATO ASI Series F, 159 (1998). Google Scholar

[20]

U. R. Freiberg, Analysis on fractal objects, Meccanica, 40 (2005), 419-436.  doi: 10.1007/s11012-005-2107-0.  Google Scholar

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F. Hausdorff, Dimension und äußeres Maß, Math. Ann., 79 (1918), 157-179.  doi: 10.1007/BF01457179.  Google Scholar

[22]

J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[23]

J. E. Keesling, he boundaries of self-similar tiles in $\mathbb{R}^n$, Topology Appl., 94 (1999), 195-205.  doi: 10.1016/S0166-8641(98)00031-5.  Google Scholar

[24]

J. E. Keesling and C. Krishnamurthi, The similarity boundary of a self similar set, Contemp. Math., 246 (1999), 175-187.  doi: 10.1090/conm/246/03783.  Google Scholar

[25]

H. Kunze, D. La Torre, F. Mendivil and E. R. Vrscay, Fractal Based Methods in Analysis, Springer, 2012. doi: 10.1007/978-1-4614-1891-7.  Google Scholar

[26]

JS. P. Lalley, Travelling salesman with self similar itenerary, Probab. Engrg. Inform. Sci., 4 (1990), 1-18.   Google Scholar

[27]

B. Mandelbort, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, 1977.  Google Scholar

[28]

M. Morán, Dynamical boundary of a self similar set, Fund. Math., 160 (1999), 1-14.   Google Scholar

[29]

P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philios. Soc., 42 (1946), 15-23.  doi: 10.1017/S0305004100022684.  Google Scholar

show all references

References:
[1]

R. K. Aswathy and S. Mathew, On different forms of self-similarity, Chaos, Solitons Fractals, 87 (2016), 102-108.  doi: 10.1016/j.chaos.2016.03.021.  Google Scholar

[2]

R. K. Aswathy and S. Mathew, Separation properties of finite products of hyperbolic iterated function systems, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 594-599.  doi: 10.1016/j.cnsns.2018.07.012.  Google Scholar

[3]

J. W. Baish and R. K. Jain, Fractals and cancer, Cancer Research, 60 (2000), 3683-3688.   Google Scholar

[4]

R. Balu and S. Mathew, On $(n, m)$-iterated function systems, Asian-Eur. J. Math., 6 (2013), 1350055, 12pp. doi: 10.1142/S1793557113500551.  Google Scholar

[5]

R. Balu, S. Mathew and N. A. Secelean, Separation properties of $(n,m)$-IFS attractors, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 160-168.  doi: 10.1016/j.cnsns.2017.04.009.  Google Scholar

[6] M. Barnsley, Fractals Everywhere, Academic Press, Boston, MA, 1988.   Google Scholar
[7]

M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A, 399 (1985), 243-275.   Google Scholar

[8]

S. S. Cross, Fractals in pathology, Journal of Pathology, 182 (1997), 1-8.  doi: 10.1002/(SICI)1096-9896(199705)182:1<1::AID-PATH808>3.0.CO;2-B.  Google Scholar

[9]

P. F. Duvall Jr. and L. S. Husch, Attractors of iterated function systems, Proc. Amer. Math. Soc., 116 (1992), 279-284.  doi: 10.1090/S0002-9939-1992-1132850-6.  Google Scholar

[10]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[11]

K. J. Falconer, Generalised dimensions of measures on self-affine sets, Nonlinearity, 12 (1999), 877-891.  doi: 10.1088/0951-7715/12/4/308.  Google Scholar

[12]

K. J. Falconer, Sub self similar sets, Trans. Amer. Math. Soc., 347 (1995), 3121-3129.  doi: 10.1090/S0002-9947-1995-1264809-X.  Google Scholar

[13]

K. J. Falconer, The dimension of self affine fractals II, Math. Proc. Cambridge Philos. Soc., 111 (1992), 169-179.  doi: 10.1017/S0305004100075253.  Google Scholar

[14]

K. J. Falconer, The Hausdorff dimension of self affine fractals, Math. Proc. Cambridge Philos. Soc., 103 (1988), 339-350.  doi: 10.1017/S0305004100064926.  Google Scholar

[15]

K. J. Falconer and J. M. Fraser, The visible part of plane self similar sets, Proc. Amer. Math. Soc., 141 (2013), 269-278.  doi: 10.1090/S0002-9939-2012-11312-7.  Google Scholar

[16]

K. J. Falconer and J. Miao, Random subsets of self affine fractals, Mathematika, 56 (2010), 61-76.  doi: 10.1112/S0025579309000357.  Google Scholar

[17]

K. J. Falconer and J. J. O'Connor, Symmetry and enumeration of self-similar fractals, Bull. Lond. Math. Soc., 39 (2007), 272-282.  doi: 10.1112/blms/bdl031.  Google Scholar

[18]

B. Forte and E. R. Vrscay, Solving the inverse problem for function/image approximation using iterated function systems I: Theoretical basis, Fractals, 2 (1994), 325-334.  doi: 10.1142/S0218348X94000429.  Google Scholar

[19]

B. Forte and E. R. Vrscay, Theory of generalized fractal transforms, Fractal Image Encoding and Analysis, NATO ASI Series F, 159 (1998). Google Scholar

[20]

U. R. Freiberg, Analysis on fractal objects, Meccanica, 40 (2005), 419-436.  doi: 10.1007/s11012-005-2107-0.  Google Scholar

[21]

F. Hausdorff, Dimension und äußeres Maß, Math. Ann., 79 (1918), 157-179.  doi: 10.1007/BF01457179.  Google Scholar

[22]

J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[23]

J. E. Keesling, he boundaries of self-similar tiles in $\mathbb{R}^n$, Topology Appl., 94 (1999), 195-205.  doi: 10.1016/S0166-8641(98)00031-5.  Google Scholar

[24]

J. E. Keesling and C. Krishnamurthi, The similarity boundary of a self similar set, Contemp. Math., 246 (1999), 175-187.  doi: 10.1090/conm/246/03783.  Google Scholar

[25]

H. Kunze, D. La Torre, F. Mendivil and E. R. Vrscay, Fractal Based Methods in Analysis, Springer, 2012. doi: 10.1007/978-1-4614-1891-7.  Google Scholar

[26]

JS. P. Lalley, Travelling salesman with self similar itenerary, Probab. Engrg. Inform. Sci., 4 (1990), 1-18.   Google Scholar

[27]

B. Mandelbort, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, 1977.  Google Scholar

[28]

M. Morán, Dynamical boundary of a self similar set, Fund. Math., 160 (1999), 1-14.   Google Scholar

[29]

P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philios. Soc., 42 (1946), 15-23.  doi: 10.1017/S0305004100022684.  Google Scholar

Figure 1.  Attractor of the product IFS $ \Phi\times \Psi. $
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Figure 2.  Attractor of the product IFS $ \Phi\times \Psi. $
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Figure 3.  Attractor of the product IFS $ \Phi\times \Psi. $
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Figure 4.  Attractor of the product IFS $ \Phi\times \Psi. $
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