On Relative Metric Mean Dimension with Potential and Variational Principles

In this article, we introduce a notion of relative mean metric dimension with potential for factor map \(\pi : (X,d, T)\rightarrow (Y, S)\) between two topological dynamical systems. To link it ergodic theory, establish four variational principles in terms entropy partitions, Shapira’s entropy, Katok’s and Brin–Katok local respectively. Some results on respect to fixed open cover are obtained the case. We also answer an question raised by Shi (On dimension, 2021. arXiv:2101.02610) partially very well-partitionable compact space, general obtain inequality involving box space. Corresponding inner given invariant measure S) investigated.