2021/01/25 by Wu, Weisheng
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2101.09934
In this article, we introduce a notion of relative mean metric dimension with potential for a factor map π: (X,d, T)→ (Y, S) between two topological dynamical systems. To link it with ergodic theory, we establish four variational principles in terms of metric entropy of partitions, Shapira's entropy, Katok's entropy and Brin-Katok local entropy respectively. Some results on local entropy with respect to a fixed open cover are obtained in the relative case. We also answer an open question raised by Shi \citeShi partially for a very well-partitionable compact metric space, and in general we obtain a variational inequality involving box dimension of the space. Corresponding inner variational principles given an invariant measure of (Y,S) are also investigated.