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This paper attempts to study the irreducibility on complete prefix code (CPC-irreducibility) of a Markov shift over a free group, a topological mixing property first considered for shift spaces over free semigroups that induces chaotic behavior such as the existence of a dense set of periodic points. An example shows that the (d , c) -reduction, an effective algorithm of determination of CPC-irreducibility of Markov shifts over free semigroups (Ban et al. in J Stat Phys 177:1043–1062, 2019), fails for general Markov shifts over free groups. This paper reveals an algorithm for determining the CPC-irreducibility of Markov shifts over both free semigroups and groups. Furthermore, such an examination is finitely checkable, and an upper bound for the complexity of the algorithm is provided. [ABSTRACT FROM AUTHOR]

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]

When a symbol or a type has been "frozen" (namely, a type of which an individual only produces one individual of the same type), its spread pattern will be changed and this change will affect the long-term behavior of the whole system. However, in a frozen system, the ξ -matrix and the offspring mean matrix are no longer primitive so that the Perron–Frobenius theorem cannot be applied directly when predicting the spread rates. In this paper, our goal is to characterize these key matrices and analyze the spread rate under more general settings both in the topological and random spread models with frozen symbols. More specifically, we propose an algorithm for explicitly computing the spread rate and relate the rate with the eigenvectors of the ξ -matrix or offspring mean matrix. In addition, we reveal that the growth of the population is exponential and that the composition of the population is asymptotically periodic. Furthermore, numerical experiments are provided as supporting evidence for the theory. [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 focuses on the analysis of two particular models, from deterministic and random perspective respectively, for spreading processes. With a proper encoding of propagation patterns, the spread rate of each pattern is discussed for both models by virtue of the substitution dynamical systems and branching process. In view of this, we are empowered to draw a comparison between two spreading processes according to their spreading models, based on which explanations are proposed on a higher frequency of a pattern in one model than the other. These results are then supported by the numerical evidence later in the article. [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 article aims to compare the long-term behavior of a spread model before and after a type in the model becomes frozen, namely, a type of which an individual only produces individuals of the same type. By means of substitution dynamical systems and matrix analysis, the first result of this work gives the spread rates of a 1-spread model with one frozen symbol. Later, in the work, this is shown to hold under more general settings, which include generalized frozen symbols and frozen symbols in m -spread models. Numerical experiments are provided as supporting evidence for the theory. [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]

In this article, we characterize the various kinds of mixing properties, in the sense of classical and complete prefix code (CPC for short) considerations, of the axial product of Z -shifts on a free semigroup G. Axial product space is an anisotropic Markov system which plays an essential role in the research of statistical physics. We reveal matrix criteria for examining these properties. Furthermore, the existence of dense (CPC-)periodic points is also demonstrated. Combining this with the aforementioned results of the (CPC-)mixing properties exhibits whether a referred system is CPC-chaotic in the sense of Devaney. [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]

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]

We consider the entropy dimension of G-shifts of finite type for the case where G is a non-Abelian monoid. Entropy dimension tells us whether a shift space has zero topological entropy. Suppose the Cayley graph CG of G has a finite representation (that is, {CgG : g ∈ G} is a finite set up to graph isomorphism), and relations among generators of G are determined by a matrix A. We reveal an association between the characteristic polynomial of A and the finite representation of the Cayley graph. After introducing an algorithm for the computation of the entropy dimension, the set of entropy dimensions is related to a collection of matrices in which the sum of each row of every matrix is bounded by the number of leaves of the graph. Furthermore, the algorithm extends to G having finitely many free-followers. [ABSTRACT FROM AUTHOR]

It has been demonstrated that excitable media with a tree structure performed better than other network topologies, therefore 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 characterization of entropy spectrum of neural networks on Cayley trees and reveals whether the entropy bifurcates when the coupling weights change. [ABSTRACT FROM AUTHOR]

We reveal an algorithm for determining the complete prefix code irreducibility (CPC-irreducibility) of dyadic trees labeled by a finite alphabet. By introducing an extended directed graph representation of tree shift of finite type (TSFT), we show that the CPC-irreducibility of TSFTs is related to the connectivity of its graph representation, which is a similar result to one-dimensional shifts of finite type. [ABSTRACT FROM AUTHOR]

This paper aims to characterize whether a multi-layer cellular neural network is of deep architecture; namely, when can an n -layer cellular neural network be replaced by an m -layer cellular neural network for m < n yet still preserve the same output phenomena? From a mathematical point of view, such characterization involves investigating whether the topological structure of two (or multiple) layers is conjugate. A decision procedure that addresses the necessary and sufficient condition for the topological conjugacy between two layers in a network is revealed. [ABSTRACT FROM AUTHOR]

This paper studies the initial value problem of multi-layer cellular neural networks. We demonstrate that the mosaic solutions of such system is topologically conjugated to a new class in symbolic dynamical systems called the path set (Abram and Lagarias in Adv Appl Math 56:109-134, ). The topological entropies of the solution, output, and hidden spaces of a multi-layer cellular neural network with initial condition are formulated explicitly. Also, a sufficient condition for whether the mosaic solution space of a multi-layer cellular neural network is independent of initial conditions is addressed. Furthermore, two spaces exhibit identical topological entropy if and only if they are finitely equivalent. [ABSTRACT FROM AUTHOR]

This paper investigates whether the output space of a multi-layer cellular neural network can be realized via a single layer cellular neural network in the sense of the existence of finite-to-one map from one output space to the other. Whenever such realization exists, the phenomena exhibited in the output space of the revealed single layer cellular neural network is at most a constant multiple of the phenomena exhibited in the output space of the original multi-layer cellular neural network. Meanwhile, the computation complexity of a single layer system is much less than the complexity of a multi-layer system. Namely, one can trade the precision of the results for the execution time. We remark that a routine extension of the proposed methodology in this paper can be applied to the substitution of hidden spaces although the detailed illustration is omitted. [ABSTRACT FROM AUTHOR]

Abstract: This manuscript considers the learning problem of multi-layer neural networks (MNNs) with an activation function which comes from cellular neural networks. A systematic investigation of the partition of the parameter space is provided. Furthermore, the recursive formula of the transition matrix of an MNN is obtained. By implementing the well-developed tools in the symbolic dynamical systems, the topological entropy of an MNN can be computed explicitly. A novel phenomenon, the asymmetry of a topological diagram that was seen in Ban, Chang, Lin, and Lin (2009) [J. Differential Equations 246, pp. 552–580, 2009], is revealed. [Copyright &y& Elsevier]

Abstract: This investigation considers the complexity of output spaces of multi-layer cellular neural networks. Let be a set of admissible local output patterns coupled with input and let be the set of admissible output patterns extracting from . Since topological entropy is an indicator for investigating the complexity of spaces, we study the topological entropy of output spaces and which are induced by and , respectively. A system has a diamond if . Necessary and sufficient conditions for the existence of diamond are demonstrated separately. Furthermore, numerical experiments exhibit some novel phenomena. [Copyright &y& Elsevier]

Abstract: The present investigation elucidates how the number of layers/variance of templates influences the phenomena of multi-layer cellular neural networks (MCNNs). This study relates to learning problems for MCNNs. We show that the greater the number of templates that MCNNs adopt, the richer the phenomena that are derived, while equivalently, such neural networks are more efficient as regards the learning aspect. Additionally, the MCNNs with more layers exhibit more phenomena than the ones with fewer layers. A novel phenomenon is seen in the study of the effect of the number of layers with respect to fixed templates. [Copyright &y& Elsevier]

Abstract: Let be the mosaic solution space of an n-layer cellular neural network. We decouple Y into n subspaces, say , and give a necessary and sufficient condition for the existence of factor maps between them. In such a case, is a sofic shift for . This investigation is equivalent to study the existence of factor maps between two sofic shifts. Moreover, we investigate whether and are topological conjugate, strongly shift equivalent, shift equivalent, or finitely equivalent via the well-developed theory in symbolic dynamical systems. This clarifies, in a multi-layer cellular neural network, each layerʼs structure. As an extension, we can decouple Y into arbitrary k-subspaces, where , and demonstrates each subspaceʼs structure. [Copyright &y& Elsevier]

Letting $ \pi:X\rightarrow Y$ be an almost-additive potential function on $ X,$ has diamond, then the pressure $ P(X,\Phi)$ $ P(Y,\pi\Phi)$ $ \rho(\Phi)=P(X,\Phi)/P(Y,\pi\Phi)$1$ --> $ \rho(\Phi)>1$ $ \left\{ (\pi_{i},X_{i})\right\} _{i=1}^{k}$ $ \pi_{i}:X_{i} \rightarrow Y$ and $ Y$ $ X_{i}\subseteq X$ $ \rho(\pi_{i},\Phi\vert _{X_{i}})$ $ P(X_{i},\Phi \vert _{X_{i}})$ $ P(Y,\pi\Phi)$ $ [1,\rho(\Phi)]$ [ABSTRACT FROM AUTHOR]

This work investigates the monotonicity of topological entropy for one-dimensional multilayer cellular neural networks. The interacting radius and number of layers are treated as parameters. Fix either one of them; the set of topological entropies grows as a strictly nested sequence with respect to one another. Apart from the comparison of the set of topological entropies, maximal and minimal templates are indicators of a dynamical system. Our results demonstrate that maximal and minimal templates of larger interacting radius (respectively number of layers) dominate those of smaller one. To be precise, the strict monotonicity of topological entropy is demonstrated through the comparison of the maximal and minimal templates as the parameters are varied. [ABSTRACT FROM AUTHOR]

This work investigates the dissipative dynamical system in the infinite lattice . The dynamics of each node depends on itself and nearby nodes by a nonlinear function. When each node is perturbed with weighted Gaussian white noise, a unique pullback attractor and forward attractor exists whose domain of attraction are random tempered sets. Furthermore, we prove that the pullback and forward attractors are equivalent to a random equilibrium which is also tempered. Both convergence to the pullback and forward attractors are exponentially fast. [ABSTRACT FROM AUTHOR]

Abstract: This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input. [Copyright &y& Elsevier]

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]

This work investigates three-dimensional pattern generation problems and their applications to three-dimensional Cellular Neural Networks (3DCNN). An ordering matrix for the set of all local patterns is established to derive a recursive formula for the ordering matrix of a larger finite lattice. For a given admissible set of local patterns, the transition matrix is defined and the recursive formula of high order transition matrix is presented. Then, the spatial entropy is obtained by computing the maximum eigenvalues of a sequence of transition matrices. The connecting operators are used to verify the positivity of the spatial entropy, which is important in determining the complexity of the set of admissible global patterns. The results are useful in studying a set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks. [ABSTRACT FROM AUTHOR]

This study demonstrates the devil's staircase structure of topological entropy functions for onedimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos. [ABSTRACT FROM AUTHOR]

This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function. [ABSTRACT FROM AUTHOR]

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function. [ABSTRACT FROM AUTHOR]

This work investigates mosaic patterns for the one-dimensional cellular neural networks with various boundary conditions. These patterns can be formed by combining the basic patterns. The parameter space is partitioned so that the existence of basic patterns can be determined for each parameter region. The mosaic patterns can then be completely characterized through formulating suitable transition matrices and boundary-pattern matrices. These matrices generate the patterns for the interior cells from the basic patterns and indicate the feasible patterns for the boundary cells. As an illustration, we elaborate on the cellular neural networks with a general 1 × 3 template. The exact number of mosaic patterns will be computed for the system with the Dirichlet, Neumann and periodic boundary conditions respectively. The idea in this study can be extended to other one-dimensional lattice systems with finite-range interaction. [ABSTRACT FROM AUTHOR]