We study variational principles for metric mean dimension. First we prove that in the variational principle of Lindenstrauss and Tsukamoto it suffices to take supremum over ergodic measures. Second we derive a variational principle for metric mean dimension involving growth rates of measure-theoretic entropy of partitions decreasing in diameter which holds in full generality and in particular does not necessitate the assumption of tame growth of covering numbers. The expressions involved are a dynamical version of Renyi information dimension. Third we derive a new expression for Geiger-Koch information dimension rate for ergodic shift-invariant measures. Finally we develop a lower bound for metric mean dimension in terms of Brin-Katok local entropy.

翻译：首先,我们证明,在林登斯特劳斯和津本的变异原则中,光是采用超模量的衡量尺度就足够了。其次,我们为直径下降的分区的测量-理论酶增速的衡量-理论酶增速的衡量平均值的变异性原则得出了一个变异性原则,这种变异性完全具有一般性,尤其不需要假定覆盖数字的纹理增长。所涉及的表达方式是Renyi信息层面的动态版本。第三,我们为Geriger-Koch信息维度的ergodid变异性措施提出了一个新的表达方式。最后,我们为Brin-Katok本地的昆虫开发了一个较低的衡量平均值约束范围。