On intrinsic invariance of fractal dimensions (Q1900080)

In this paper a fractal dimension function \(\dim_F\) is defined as a function which assigns to a metric space \((X, \rho)\) a real number at least \(- 1\) and satisfies the axioms of Lipschitz subinvariance, of stability and of equality to the topological dimension for smooth submanifolds \(X\) of \(\mathbb{R}^n\). The authors consider geometrically acceptable metric spaces \((X, \rho)\) introduced by Borsuk, i.e., they are strongly arcwise connected and the intrinsic metric \(\rho^*\) is topologically equivalent to \(\rho\). Metrics will be compared as follows. If \(\rho_1, \rho_2\) are metrics for \(X\) then \(\rho_1 \prec \rho_2\) iff \(\exists \alpha > 0 : \rho_1 \leq \alpha \rho_2\). An intrinsic isometry on the spaces \(X\) and \(Y\) is an isometry with respect to \(\rho^*_X, \rho^*_Y\). The authors show that in the general setting intrinsic isometries do not preserve Hausdorff as well as box dimension, both are examples for \(\dim_F\). They give also a sufficient condition for dimension invariance with respect to such isometries. The key is the above order for the metrics.