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February  2022, 15(2): 265-281. doi: 10.3934/dcdss.2021004

On properties of similarity boundary of attractors in product dynamical systems

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

* Corresponding author: Nitha Niralda P C

Received  April 2020 Revised  October 2020 Published  February 2022 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.

Citation: Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 265-281. 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.

[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.

[3]

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

[4]

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

[5]

R. BaluS. 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.

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

[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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[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.

[16]

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

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[29]

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

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.

[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.

[3]

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

[4]

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

[5]

R. BaluS. 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.

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

[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.

[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.

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[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.

[16]

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

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[29]

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

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