Some Notes on Variational Principle for Metric Mean Dimension

In this note, we establish variational principles for mean dimension and metric mean dimension. Our main results are as follows. (1) We establish a variational principle for metric mean dimension in terms of
$L^{\infty }$
rate-distortion function with supremum over all ergodic measures, which answers a question posed by Gutman and Śpiewak in (Around the variational principle for metric mean dimension, Studia Math. (2021) 261 345-60). (2) We establish a double variational principle for mean dimension in terms of mean Rényi information dimension for the systems admitting marker property. (3) If the system admits marker property, we show that the order of sup and limsup of the variational principle for metric mean dimension in terms of Rényi information dimension introduced by Gutman and Śpiewak (2021) can be exchanged for some “nice” metrics.