Topological Entropy Dimension and Directional Entropy Dimension for Z 2 -Subshifts

Abstract

The notion of topological entropy dimension for a Z -action has been introduced to measure the subexponential complexity of zero entropy systems. Given a Z 2 -action, along with a Z 2 -entropy dimension, we also consider a finer notion of directional entropy dimension arising from its subactions. The entropy dimension of a Z 2 -action and the directional entropy dimensions of its subactions satisfy certain inequalities. We present several constructions of strictly ergodic Z 2 -subshifts of positive entropy dimension with diverse properties of their subgroup actions. In particular, we show that there is a Z 2 -subshift of full dimension in which every direction has entropy 0.

Keywords

• ℤ2-action
• entropy dimension
• directional entropy dimension
• entropy generating shape
• minimal