The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight for analyzing entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate joinings among entropy zero systems and prove the disjointness property among some classes of entropy zero systems using dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.

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

For every infinite (countable discrete) amenable group G and every positive integer d we construct a minimal G-action of mean dimension d/2 which cannot be embedded in the full G-shift on ([0,1](d))(G) .

Entropy dimension is an entropy-type quantity which takes values in and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight to analyze the entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate the joinings among entropy zero systems and prove the disjointness property among entropy zero systems using the dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.