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.

We generalize the notion of reversible systems in symbolic dynamics, and investigate their properties. It is shown that a reversible topological Markov shift can be represented by a pair of matrices of special types. This enables us to classify the invariant measures of the reversible systems. Necessary and/or sufficient conditions for the existence of a reversal of finite order are established in terms of the adjacency matrices. We also prove that a topological Markov shift with a reversal of order two admits reversals of all orders.

Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$ lie in the closed upper half plane $\rm{Im}\ z\geq 0$, or if $f_1, f_2, \dots$ are real polynomials and the numbers of their non-real zeros are uniformly bounded, then the sequence converges uniformly on compact sets in the complex plane. The results imply a theorem of Benz and a conjecture of P\'olya.

The notion of entropy dimension has been introduced to measure the subexponential complexity of zero entropy systems. In this work we present a general construction of a strictly ergodic subshift of topological entropy dimension $\alpha$ for each $\alpha \in (0,1)$. It is shown that the system satisfies some sort of regularity in the size of atoms and the first return time. Moreover, we modify the construction to obtain a variant system that is weakly mixing.