Remarks on the Invariance Theorem of Generalized Fractal Dimensions of Graphs of Continuous Functions

This paper establishes a comprehensive theory for the behavior of the generalized box, Hausdorff, and packing dimensions under composition of continuous functions. We develop two complementary approaches: a measure-theoretic framework based on energy integrals and potential theory for analyzing the generalized Hausdorff and packing dimensions, and a covering-based approach for the generalized box dimensions. Our main results characterize how generalized fractal dimensions of graphs of functions transform under composition with Lipschitz and bi-Lipschitz functions. For outer function variation, it has been shown that composition with Lipschitz functions does not increase dimensions, while bi-Lipschitz functions preserve dimensions exactly. For inner function variation, distinct results for different types of dimensions have been established. For the generalized box dimensions, Lipschitz inner functions yield dimensional variance, while bi-Lipschitz inner functions ensure invariance. For the generalized Hausdorff and packing dimensions, bi-Lipschitz inner functions are required for dimensional invariance. Applications include dimensional invariance for compositions with elementary functions such as power functions, roots, reciprocals, exponentials, and logarithms. Explicit examples of generalized fractal dimensions satisfying our framework have also been provided. Furthermore, we study the effect of the Hölder and counter-Hölder continuity on the generalized lower box dimension of graphs of functions, establishing an exact value [Formula: see text] for functions that are simultaneously [Formula: see text]-order Hölder and counter-Hölder continuous.