We study Markov tree-shifts given by $k$ transition matrices, one for each of its $k$ directions. We provide a method to characterize the complexity function for these tree-shifts, used to calculate the tree entropies defined by Ban and Chang arXiv:1509.08325 and Petersen and Salama arXiv:1712.02251. Moreover, we compare these definitions of entropy in order to determine some of their properties. The characterization of the complexity function provided is used to calculate the entropy of some examples. The question of existence of a specific type of invariant measures for such tree-shifts is addressed. Finally, we analyse some topological properties introduced by Ban and Chang arXiv:1509.01355 for the purpose of answering two of the questions raised by these authors.
We give a definition of topological entropy for tree shifts, prove that the limit in the definition exists, and show that it dominates the topological entropy of the associated one-dimensional shift of finite type when the labeling of the tree shares the same restrictions.
It has been demonstrated that excitable media with a tree structure performed better than other network topologies, it is natural to consider neural networks defined on Cayley trees. The investigation of a symbolic space called tree-shift of finite type is important when it comes to the discussion of the equilibrium solutions of neural networks on Cayley trees. Entropy is a frequently used invariant for measuring the complexity of a system, and constant entropy for an open set of coupling weights between neurons means that the specific network is stable. This paper gives a complete characteriz…
We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not a conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In doi:10.1016/j.tcs.2018.05.034 and doi:10.3934/dcds.2020186, Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and $h(\mathcal{T}_X) \geq h(X)$, where $\mathcal{T}_X$ is the hom tree-shift derived from $X$. We characterize a necessary and sufficient condition when the equality holds for the case where $X$ is a shift of finite type. In addition, two…
I give a simple proof for the fact that positive entropy subshifts contain infinite binary trees where branching happens synchronously in each branch, and that the branching times form a set with positive lower asymptotic density.
We consider the existence of the topological entropy of shift spaces on a finitely generated semigroup whose Cayley graph is a tree. The considered semigroups include free groups. On the other hand, the notion of stem entropy is introduced. For shift spaces on a strict free semigroup, the stem entropy coincides with the topological entropy. We reveal a sufficient condition for the existence of the stem entropy of shift spaces on a semigroup. Furthermore, we demonstrate that the topological entropy exists in many cases and is identical to the stem entropy.
We prove that a zero topological entropy continuous tree map always displays zero topological sequence entropy when it is restricted to its non-wandering and chain recurrent sets. In addition, we show that a similar result is not possible when the phase space is a dendrite even when we consider only the restriction on the set of periodic points.
In this paper, we first concentrate on the possible values and dense property of entropies for isotropic and anisotropic axial products of subshifts of finite type (SFTs) on $\mathbb{N}^d$ and $d$-tree $\mathcal{T}_d$. We prove that the entropies of isotropic and anisotropic axial products of SFTs on $\mathbb{N}^d$ are dense in $[0,\infty)$, and the same result also holds for anisotropic axial products of SFTs on $\mathcal{T}_d$. However, the result is no longer true for isotropic axial products of SFTs on $\mathcal{T}_d$. Next, motivated by the work of Johnson, Kass and Madden [16], and Schra…
This paper deals with the topological entropy for hom Markov shifts $\mathcal{T}_M$ on $d$-tree. If $M$ is a reducible adjacency matrix with $q$ irreducible components $M_1, \cdots, M_q$, we show that $h(\mathcal{T}_{M})=\max_{1\leq i\leq q}h(\mathcal{T}_{M_{i}})$ fails generally, and present a case study with full characterization in terms of the equality. Though that it is likely the sets $\{h(\mathcal{T}_{M}):M\text{ is binary and irreducible}\}$ and $\{h(\mathcal{T}_{X}):X\text{ is a one-sided shift}\}$ are not coincident, we show the two sets share the common closure. Despite the fact tha…
We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is $\mathbb Z$. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approx…
In a recent paper by A. Backhausz, B. Gerencsér and V. Harangi, it was shown that factors of independent identically distributed random processes (IID) on trees obey certain geometry-driven inequalities. In particular, the mutual information shared between two vertices decays exponentially, and there is an explicit bound for this decay. In this note we show that all of these inequalities could be verbatim translated to the setting of factors of processes driven by unique Gibbs measures. As a consequence, we show that correlations decay exponentially for unique Gibbs measures on trees.
