Relative maximal pattern entropy

The notion of relative maximal pattern entropy is investigated in this paper. It turns out that in both topological and measure-theoretical settings, the relative maximal pattern entropy coincides with the supremum of relative sequence entropies over all infinite increasing sequences. Furthermore, leveraging the development of the local theory for relative maximal pattern entropy, we demonstrate that for any topological dynamical system or any ergodic measure-preserving system, the relative maximal pattern entropy is precisely $ \log k $ for some $ k\in\mathbb{N}\cup \{\infty\} $ with $ k $ the superior length of essential relative maximal pattern entropy tuples.