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In this article, we introduce the concept of boundary complexity and prove that for a 2-multiplicative integer system (2-MIS) X Ω p on N (or X Ω p on N d , d ≥ 2), every point in [ h ( X Ω p ) , log ⁡ r ] can be realized as a boundary complexity of a 2-MIS at a specific speed, where r stands for the size of the alphabet. The result is new and quite different from the N d subshift of finite type (SFT) for d ≥ 1. Furthermore, the formula of surface entropy for a N d 2-MIS is also presented. This illustrates an efficient method to calculate the topological entropy for N d 2-MIS and also provides intrinsic differences between N d k -MIS and SFTs for d ≥ 1 and k ≥ 2. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the topological entropy of a q -multiplicative subshift on tree (q -MST), that is, every boundary of the tree follows the rule of some one-dimensional multiplicative integer systems defined by Kenyon et al. [20]. Supposing the number of vertices in the tree T grows exponentially, the explicit formula for the topological entropy of the q -MST is presented in Theorem 1.1. Next, we consider the coupled q -MST, namely, every boundary of T satisfies the q -multiple constraint and is characterized by subshift of finite type. The comparison of topological entropy for the coupled and non-coupled q -MSTs is provided in Theorem 1.4. Furthermore, the explicit formula of some coupled q -MSTs are also obtained herein. Finally, the entropy formula for the general q -MSTs, known as X Ω (S) , is described in Theorem 1.6. [ABSTRACT FROM AUTHOR]

This paper establishes the large deviation principle (LDP) of certain types of nonconventional ergodic averages, namely, 1 N S N ∗ and 1 N S N # on N (defined later). The LDP for both averages are presented and such a result extends the preceding work of (Carinci et al. in Indag Math 23(3):589–602, 2012) to some specific cases of d-multiple averages for d ≥ 3 . [ABSTRACT FROM AUTHOR]

This work introduces constructive and systematic methods for verifying the topological mixing and strong specification (or strong irreducibility) of two-dimensional shifts of finite type. First, we define transition matrices on infinite strips of width n for all n ≥ 2. To determine the primitivity of the transition matrices, we introduce the connecting operators that reduce the high-order transition matrices to lower-order transition matrices. Then, two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. Then, the primitivity, corner-extendability, and crisscross-extendability are used to demonstrate the topological mixing. Finally, we show that the hole-filling condition yields the strong specification property. The application of all the above-mentioned conditions can be verified in a finite number of steps. [ABSTRACT FROM AUTHOR]

A multiplicative integer subshift X Ω derived from the subshift Ω is invariant under multiplicative integer action, which is closely related to the level set of multiple ergodic average. The complexity of X Ω is usually measured by entropy (or box dimension). This work concerns on two types of multi-dimensional multiplicative integer subshifts (MMIS) with different coupling constraints, and then obtains their entropy formulae. [ABSTRACT FROM AUTHOR]

This work investigates zeta functions for d-dimensional shifts of finite type, d ≥ 3. First, the three-dimensional case is studied. The trace operator Ta1,a2;b12 and rotational matrices Rx;a1,a2;b12 and Ry;a1,a2;b12 are introduced to study ${\scriptsize\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{}} a_{1} & b_{12} & b_{23}\\[1pt] 0 & a_{2} & b_{23} \\[1pt] 0 & 0 & a_{3} \end{array}\right]}$ -periodic patterns. The rotational symmetry of Ta1,a2;b12 induces the reduced trace operator τa1,a2;b12 and then the associated zeta function ζa1,a2;b12 = (det(I-sa1a2τa1,a2;b12))-1. The zeta function ζ is then expressed as $\zeta=\prod_{a_{1}=1}^{\infty}\prod_{a_{2}=1}^{\infty} \prod_{b_{12}=0}^{a_{1}-1}\zeta_{a_{1},a_{2};b_{12}}$, a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL3(ℤ). Hence, a family of zeta functions exists with the same integer coefficients in their Taylor series expansions at the origin, and yields a family of identities in number theory. The methods used herein are also valid for d-dimensional cases, d ≥ 4, and can be applied to thermodynamic zeta functions for the three-dimensional Ising model with finite range interactions. [ABSTRACT FROM AUTHOR]

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 [16,17] , Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and h (T X) ≥ h (X) , where 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. Additionally, two novel phenomena have been revealed for tree-shifts. There is a gap in the set of topological entropy of hom tree-shifts of finite type, making such a set not dense. Last but not least, the topological entropy of a reducible hom tree-shift of finite type can be strictly larger than that of its maximal irreducible component. [ABSTRACT FROM AUTHOR]

This paper deals with the topological entropy for hom Markov shifts T M on d -tree. If M is a reducible adjacency matrix with q irreducible components M 1 , ⋯ , M q , we show that h (T M) = max 1 ≤ i ≤ q ⁡ h (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 (T M) : M is binary and irreducible } and { h (T X) : X is a one-sided shift } are not coincident, we show the two sets share the common closure. Despite the fact that such closure is proved to contain the interval [ d log ⁡ 2 , ∞) , numerical experiments suggest its complement contain open intervals. [ABSTRACT FROM AUTHOR]

The purpose of this article is twofold. On one hand, we reveal the equivalence of shift of finite type between a one-sided shift X and its associated hom tree-shift T X , as well as the equivalence in the sofic shift. On the other hand, we investigate the interrelationship among the comparable mixing properties on tree-shifts as those on multidimensional shift spaces. They include irreducibility, topologically mixing, block gluing, and strong irreducibility, all of which are defined in the spirit of classical multidimensional shift, complete prefix code (CPC), and uniform CPC. In summary, the mixing properties defined in all three manners coincide for T X. Furthermore, an equivalence between irreducibility on T A and irreducibility on X A are seen, and so is one between topologically mixing on T A and mixing property on X A , where X A is the one-sided shift space induced by the matrix A and T A is the associated tree-shift. These equivalences are consistent with the mixing properties on X or X A when viewed as a degenerate tree-shift. [ABSTRACT FROM AUTHOR]