Upper Metric Mean Dimensions with Potential

The aim of this paper is to establish a variational principle between upper metric mean dimension with potential and upper measure-theoretic mean dimension with potential, which are potentialized versions of upper metric mean dimensions introduced by Lindenstrauss and Weiss (Israel J Math 115:1-24, 2000). As a corollary, we get an alternative variational principle different from those in Lindenstrauss and Tsukamoto (Geom Flint Anal 29:1048-1109, 2019, IEEE Trans Inf Theory 64(5):3590-3609, 2018), Tsukamoto (Adv Math 361(12):106935, 2019), Velozo and Velozo (Rate distortion theory, metric mean dimension and measure theoretic entropy. https://arxiv.org/abs/1707.05762) . Moreover, we also consider local versions of upper metric mean dimensions with potential by introducing similar notions for pseudo-orbits. We further explore the relations between the local versions and global versions. At last, a few examples are investigated.