MEAN DIMENSION THEORY IN SYMBOLIC DYNAMICS FOR FINITELY GENERATED AMENABLE GROUPS

In this paper, we mainly show a close relationship between topo-logical entropy and mean dimension theory for actions of polynomial growth groups. We show that metric mean dimension and mean Hausdorff dimension of subshifts with respect to the lower rank subgroup are equal to its topological entropy multiplied by the growth rate of the subgroup. Meanwhile, we prove that above result holds for rate distortion dimension of subshifts with respect to a lower rank subgroup and measure entropy. Furthermore, we present some examples.