Characterization for entropy of shifts of finite type on Cayley trees

The notion of tree-shifts constitutes an intermediate class in between one-sided shift spaces and multidimensional ones. This paper proposes an algorithm for computing of the entropy of a tree-shift of finite type. Meanwhile, the entropy of a tree-shift of finite type is $\dfrac{1}{p} \ln \lambda$ for some $p \in \mathbb{N}$, where $\lambda$ is a Perron number. This extends Lind's work on one-dimensional shifts of finite type. As an application, the entropy minimality problem is investigated, and we obtain the necessary and sufficient condition for a tree-shift of finite type being entropy minimal with some additional conditions

On the topological pressure of axial product on trees

This article investigates the topological pressure of isotropic axial products of Markov subshift on the $d$-tree. We show that the quantity increases with dimension $d$, and demonstrate that, with the introduction of surface pressure, the two types of pressure admit the same asymptotic value. To this end, the pattern distribution vectors and the associated transition matrices are introduced herein to partially transplant the large deviation theory to tree-shifts, and so the increasing property is proved via an almost standard argument. An application of the main result to a wider class of shift spaces is also provided in this paper, and numerical experiments are included for the purpose of verification