We introduce two types of (q, μ)-invariance entropies for topological time-invariant control systems. Through an analysis of level sets of local invariance entropies, we establish relationships between Bowen invariance entropy and (q, μ)-Bowen invariance entropy, as well as between packing invariance entropy and (q, μ)-packing invariance entropy. Moreover, we provide an upper estimate on the multifractal spectrum for local invariance entropies, enhancing the understanding of complexity in control systems.
Hu et al. (Sun,) studied this question.