On the Dimensional Invariance of Graphs of Fractal Functions Under Horizontal Nonlinear Variation

On the basis of our previous work, this paper makes an investigation on whether fractal dimensions of an object with a fractal curve shape will keep invariable under being subjected to horizontal nonlinear variation in this paper. For two fractal continuous functions $\phi \left(x\right)$ and $\psi \left(x\right)$, it has been proven that the graph of $(\phi \circ \psi )\left(x\right)$ has the same lower and upper Box dimensions as the graph of $\phi \left(x\right)$ when $\psi \left(x\right)$ is a monotonic bi-Lipschitz function with certain basic elementary functions provided. Further, we show that such invariance property also holds for the Hausdorff dimension, the packing dimension, and the Hewitt–Stromberg dimension under certain conditions. Numerical simulations of some concrete examples have also been carried out to corroborate our theoretical results. This work may contribute to the dimensional theory of graphs of fractal functions and have certain practical application significance in physics and other natural sciences.