Topological Entropy Dimension and Directional Entropy Dimension for ℤ2-Subshifts

The notion of topological entropy dimension for a Z-action has been introduced to measure the subexponential complexity of zero entropy systems. Given a Z2-action, along with a Z2-entropy dimension, we also consider a finer notion of directional entropy dimension arising from its subactions. The entropy dimension of a Z2-action and the directional entropy dimensions of its subactions satisfy certain inequalities. We present several constructions of strictly ergodic Z2-subshifts of positive entropy dimension with diverse properties of their subgroup actions. In particular, we show that there is a Z2-subshift of full dimension in which every direction has entropy 0.