Tree-shifts: Irreducibility, mixing, and the chaos of tree-shifts

Abstract

Abstract. Topological behavior, such as chaos, irreducibility, and mixing of a one-sided shift of finite type, is well elucidated. Meanwhile, the investigation of multidimensional shifts, for instance, textile systems is difficult and only a few results have been obtained so far.This paper studies shifts defined on infinite trees, which are called tree-shifts. Infinite trees have a natural structure of one-sided symbolic dynamical systems equipped with multiple shift maps and constitute an intermediate class in between one-sided shifts and multidimensional shifts. We have shown not only an irreducible tree-shift of finite type, but also a mixing tree-shift that are chaotic in the sense of Devaney. Furthermore, the graph and labeled graph representations of tree-shifts are revealed so that the verification of irreducibility and mixing of a tree-shift is equivalent to determining the irreducibility and mixing of matrices, respectively. This extends the classical results of one-sided symbolic dynamics.A necessary and sufficient condition for the irreducibility and mixing of tree-shifts of finite type is demonstrated. Most important of all, the examination can be done in finite steps with an upper bound.

Tree-shifts: the entropy of tree-shifts of finite type

“…Lemma 2.2 It only remains to deal with Cases (1)- (6). Notably, Cases (1)- (3) can be treated similarly.…”

“…Readers are referred to [14] for more details. In [6], we show that every tree-shift of finite type is topological conjugate to a vertex tree-shift, which is defined analogously to the definition of matrix shifts. We conjecture that the answer to the above problem is affirmative, and the related work is in preparation.…”

“…Readers are referred to [4,5] for more details. This paper, which is a consequential investigation of [6], focuses on the entropy of tree-shifts of finite type and intends to reveal the connection between tree-shifts of finite type and nonlinear dynamical systems. We demonstrate that the characterization of entropy of tree-shifts of finite type relates to solving systems of nonlinear recurrence equations (Theorems 1.5 and 1.6).…”

Complexity of shift spaces on semigroups

“…Let n ∈ N and denote by E n the set of elements in G whose length are less than or equal to n. We define Γ n (X) the set of all possible patterns of X in E n and set γ n = |Γ n |, i.e., the number of Γ n . The topological entropy (entropy for short) of X is defined as (1) h(X) = lim sup n→∞ ln γ n |E n | .…”

Mixing properties of tree-shifts

“…In addition, we show that every TSFT is conjugate to a vertex tree-shift (defined later), which is a TSFT represented by finitely many 0-1 matrices. After illustrating that computing the topological entropy of a TSFT is equivalent to the investigation of a system of nonlinear recurrence equations [4], Akiyama et al indicate that the collection of topological entropies of TSFTs, like one-dimensional SFTs, is the set of Perron numbers [1].…”