What is the variant of fractal dimension under addition of functions with same dimension and related discussions

Ruhua Zhang; Wei Xiao; Ruhua Zhang; Wei Xiao

doi:10.3934/math.2024938

AIMS Mathematics

2024, Volume 9, Issue 7: 19261-19275. doi: 10.3934/math.2024938

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Research article

What is the variant of fractal dimension under addition of functions with same dimension and related discussions

Ruhua Zhang ^,,
Wei Xiao

School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, China

Received: 09 January 2024 Revised: 07 April 2024 Accepted: 19 April 2024 Published: 11 June 2024
MSC : 26A33, 28A80

This paper attempts to explore possible box dimension of two added fractal continuous functions with the same dimension. Two interesting and meaningful results are obtained. Let $ g(x) $ and $ h(x) $ have the same box dimension $ t\; (1 < t\leq 2) $, the box dimension of $ g(x)+h(x) $ may or may not exist. If it exists, it can take an arbitrary real number $ \gamma $ satisfying $ 1 < \gamma\leq t $. If it does not exist, its lower and upper box dimensions can reach arbitrary different real numbers $ t_1\; {\rm and}\; t_2 $ that satisfy $ 1 < t_1 < t_2 < t\leq 2 $. These unexpected conclusions drive us to probe into the characteristics of collection of all fractal continuous functions with the same box dimension under ordinary linear operations (scalar multiplication and addition). Following the known fractal features of some typical fractal functions such as the Weierstrass function $ W_t(x) $, we classify the fractal functions into three types: consistent fractal functions, non-consistent fractal functions, and simple fractal functions. By utilizing these classifications and fractal feature descriptions, the causality of the box dimension of two added fractal functions can be partially revealed. We hope that these initial superficial discussions will lead deeper consideration on the essence of variants of fractal dimension under linear combinations of fractal functions. Moreover, these fractal features may be applied further in other fields of fractals.
- box dimension,
- fractal continuous function,
- summation of two fractal functions
Citation: Ruhua Zhang, Wei Xiao. What is the variant of fractal dimension under addition of functions with same dimension and related discussions[J]. AIMS Mathematics, 2024, 9(7): 19261-19275. doi: 10.3934/math.2024938

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Abstract

This paper attempts to explore possible box dimension of two added fractal continuous functions with the same dimension. Two interesting and meaningful results are obtained. Let $ g(x) $ and $ h(x) $ have the same box dimension $ t\; (1 < t\leq 2) $, the box dimension of $ g(x)+h(x) $ may or may not exist. If it exists, it can take an arbitrary real number $ \gamma $ satisfying $ 1 < \gamma\leq t $. If it does not exist, its lower and upper box dimensions can reach arbitrary different real numbers $ t_1\; {\rm and}\; t_2 $ that satisfy $ 1 < t_1 < t_2 < t\leq 2 $. These unexpected conclusions drive us to probe into the characteristics of collection of all fractal continuous functions with the same box dimension under ordinary linear operations (scalar multiplication and addition). Following the known fractal features of some typical fractal functions such as the Weierstrass function $ W_t(x) $, we classify the fractal functions into three types: consistent fractal functions, non-consistent fractal functions, and simple fractal functions. By utilizing these classifications and fractal feature descriptions, the causality of the box dimension of two added fractal functions can be partially revealed. We hope that these initial superficial discussions will lead deeper consideration on the essence of variants of fractal dimension under linear combinations of fractal functions. Moreover, these fractal features may be applied further in other fields of fractals.

References

[1]	W. Xiao, Cardinality and fractal linear subspace about fractal functions, Fractals, 30 (2022), 2250146. https://doi.org/10.1142/S0218348X22501468 doi: 10.1142/S0218348X22501468
[2]	A. S. Besicovitch, H. D. Ursell, Sets of fractional dimensions (Ⅴ): On dimensional numbers of some continuous curves, J. London Math. Soc., s1-12 (1937), 18–25. https://doi.org/10.1112/jlms/s1-12.45.18 doi: 10.1112/jlms/s1-12.45.18
[3]	T. Bedford, The box dimension of self-affine graphs and repellers, Nonlinearity, 2 (1989), 53. https://doi.org/10.1088/0951-7715/2/1/005 doi: 10.1088/0951-7715/2/1/005
[4]	K. Falconer, Fractal geometry: Mathematical foundations and applications, Chichester, West Sussex: Wiley, 2003. https://doi.org/10.1002/0470013850
[5]	Z. Y. Wen, Mathematical foundations of fractal geometry (in Chinese), Shanghai: Science Technology Education Publication House, 2000.
[6]	R. R. Nigmatullin, Fractional integral and its physical interpretation, Theor. Math. Phys., 90 (1992), 242–251. https://doi.org/10.1007/BF01036529 doi: 10.1007/BF01036529
[7]	F. Y. Ren, Z. G. Yu, F. Su, Fractional integral associated to the self-similar set or the generalized self-similar set and its physical interpretation, Phys. Lett. A, 219 (1996), 59–68. https://doi.org/10.1016/0375-9601(96)00418-5 doi: 10.1016/0375-9601(96)00418-5
[8]	K. Barański, On the dimension of graphs of Weierstrass-type functions with rapidly growing frequencies, Nonlinearity, 25 (2012), 193. https://doi.org/10.1088/0951-7715/25/1/193 doi: 10.1088/0951-7715/25/1/193
[9]	M. V. Berry, Z. V. Lewis, On the Weierstrass-Mandelbrot fractal function, Proc. R. Soc. London A, 370 (1980), 459–484. https://doi.org/10.1098/rspa.1980.0044 doi: 10.1098/rspa.1980.0044
[10]	B. R. Hunt, The Hausdorff dimension of graphs of Weierstrass functions, Proc. Amer. Math. Soc., 126 (1998), 791–800. https://doi.org/10.1090/S0002-9939-98-04387-1 doi: 10.1090/S0002-9939-98-04387-1
[11]	D. C. Sun, Z. Y. Wen, Dimension de Hausdorff des graphs de s$\rm\acute{e}$ries trigonom$\rm\acute{e}$triques laculaires, C. R. Acad. Sci. Paris S$\rm\acute{e}$rie I. Math., 310 (1990), 135–140.
[12]	D. C. Sun, Z. Y. Wen, The Hausdorff dimension of graphs of a class of Weierstrass functions, Prog. Nat. Sci., 6 (1996), 547–553.
[13]	T. F. Xie, S. P. Zhou, On a class of singular continuous functions with graph Hausdorff dimension 2, Chaos Soliton Fract., 32 (2007), 1625–1630. https://doi.org/10.1016/j.chaos.2005.12.038 doi: 10.1016/j.chaos.2005.12.038
[14]	X. X. Cui, W. Xiao, What is the effect of the Weyl fractional integral on the Hölder continuous functions? Fractals, 29 (2021), 2150026. https://doi.org/10.1142/S0218348X21500262 doi: 10.1142/S0218348X21500262
[15]	Y. S. Liang, W. Y. Su, Riemann-Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math., 46 (2016), 423–438. https://doi.org/10.1360/012016-1 doi: 10.1360/012016-1
[16]	W. Xiao, Relationship of upper box dimension between continuous fractal functions and their Riemann-Liouville fractional integral, Fractals, 29 (2021), 2150264. https://doi.org/10.1142/S0218348X21502649 doi: 10.1142/S0218348X21502649
[17]	W. Xiao, Y. S. Liang, On Riemann-Liouville fractional differentiability of continuous functions and its physical interpretation, Fractals, 29 (2021), 2150242. https://doi.org/10.1142/S0218348X2150242X doi: 10.1142/S0218348X2150242X
[18]	W. Xiao, On box dimension of Hadamard fractional integral (Partly answer fractal calculus conjecture), Fractals, 30 (2022), 2250094. https://doi.org/10.1142/S0218348X22500943 doi: 10.1142/S0218348X22500943
[19]	W. X. Shen, Hausdorff dimension of the graphs of the classical Weierstrass functions, Math. Z., 289 (2018), 223–266. https://doi.org/10.1007/s00209-017-1949-1 doi: 10.1007/s00209-017-1949-1
[20]	J. R. Wu, The effects of the Riemann-Liouville fractional integral on the Box dimension of fractal graphs of hölder continuous functions, Fractals, 28 (2020), 2050052. https://doi.org/10.1142/S0218348X20500528 doi: 10.1142/S0218348X20500528
[21]	Y. S. Liang, Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions, Fract. Calc. Appl. Anal., 21 (2018), 1651–1658. https://doi.org/10.1515/fca-2018-0087 doi: 10.1515/fca-2018-0087
[22]	J. Wang, K. Yao, Dimension analysis of continuous functions with unbounded variation, Fractals, 25 (2017), 1730001. https://doi.org/10.1142/S0218348X1730001X doi: 10.1142/S0218348X1730001X
[23]	H. J. Ruan, W. Y. Su, K. Yao, Box dimension and fractional integral of linear fractal interpolation functions, J. Approx. Theory, 161 (2009), 187–197. https://doi.org/10.1016/j.jat.2008.08.012 doi: 10.1016/j.jat.2008.08.012
[24]	Q. Zhang, Some remarks on one-dimensional functions and their Riemann-Liouville fractional calculus, Acta. Math. Sin.-English Ser., 30 (2014), 517–524. https://doi.org/10.1007/s10114-013-2044-0 doi: 10.1007/s10114-013-2044-0
[25]	Y. S. Liang, Definition and classification of one-dimensional continuous functions with unbounded variation, Fractals, 25 (2017), 1750048. https://doi.org/10.1142/S0218348X17500487 doi: 10.1142/S0218348X17500487
[26]	B. B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Company, 1982.

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