Shannon Entropy and Fractal Dimensions: An Asymptotic Entropic Quotient and Its Finite-Scale Estimation

This study investigates conditions under which information dimension provides an upper bound for Hausdorff dimension using Shannon entropy and box occupancy. We establish a sufficient entropic condition and introduce the asymptotic entropic quotient, which measures the proportion of maximal entropy retained across scales. Its complement quantifies entropy loss, while its reciprocal defines an entropic correction factor. Cantor-type examples show that the quotient distinguishes distributions sharing the same geometric support. For finite samples, we justify a constrained least-squares estimator and develop a multi-criteria procedure for selecting the principal scaling window. The framework is applied to the Hénon, Lorenz, Ikeda, Rössler, and Chua double-scroll attractors, with sensitivity analyses for sample size and window selection. Estimated entropy losses range from 3.69% to 8.35%, and the entropic quotients remain stable across the tested conditions. The correction improves agreement with the reported Hausdorff dimension for Lorenz. For Hénon, it does not improve that agreement, although both estimates remain compatible with reported fractal dimensions. Results for Ikeda, Rössler, and Chua are consistent in scale with reported dimensions. Overall, the quotient provides distributional information beyond box occupancy and supports a finite-scale interpretation of entropy loss and its associated dimensional correction.