RESEARCH ON FRACTAL DIMENSIONS AND THE HÖLDER CONTINUITY OF FRACTAL FUNCTIONS UNDER OPERATIONS

Binyan Yu and Yongshun Liang ()
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Binyan Yu: School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China
Yongshun Liang: School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China

FRACTALS (fractals), 2024, vol. 32, issue 03, 1-28

Abstract: Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.

Keywords: Fractal Dimensions; The Hölder Continuity; Graph of the Function; Continuous Functions; The Operation Structure (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S0218348X2450052X

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