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Jung-Chao Ban

Publications and source records attributed to Jung-Chao Ban.

At least 19 recordsLinked to original sources
arXiv2025

Hausdorff dimensions of Beatty multiple shifts

In this paper, the Beatty multiple shift is introduced, which is a generalization of the multiplicative shift of finite type (multiple SFT) [Kenyon, Peres and Solomyak, Ergodic Theory and Dynamical Systems, 2012] and the affine multiple shift [Ban, Hu, Lai and Liao, Advances in Mathematics, 2025]. The Hausdorff and Minkowski dimension formulas are obtained, and the coefficients of the formula is closely related to the classical disjoint covering of the positive integers in number theory.

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arXiv2025

Hausdorff dimensions of irreducible Markov hom tree-shifts

This paper features a Cramér's theorem for finite-state Markov chains indexed by rooted $d$-trees, obtained via the method of types in the classical analysis of large deviations. Along with the theorem comes two applications: an almost-sure type convergence of sample means and a formula for the Hausdorff dimension of the symbolic space associated with the irreducible Markov chain.

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arXiv2025

On the topological pressure of axial product on trees

This article investigates the topological pressure of isotropic axial products of Markov subshifts on the $d$-tree. We show that the quantity increases with dimension $d$. To achieve this, we introduce the pattern distribution vectors and the associated transition matrices and partially transplant the large deviation theory to tree-shifts. Additionally, we apply our main result to a broader class of shift spaces, accompanied by numerical experiments for verification.

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arXiv2025

Projected Spread Models

We present a disease transmission model that considers both explicit and non-explicit factors. This approach is crucial for accurate prediction and control of infectious disease spread. In this paper, we extend the spread model from our previous works \cite{ban2021mathematical,ban2023randomspread, ban2023mathematical, ban2023spread} to a projected spread model that considers both hidden and explicit types. Additionally, we provide the spread rate for the projected spread model corresponding to the topological and random models. Furthermore, examples and numerical results are provided to illustrate the theory.

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arXiv2024

On the independence of shifts defined on $\mathbb{N}^d$ and trees

In this paper, we study the independence of shifts defined on $\mathbb{N}^d$ ($\mathbb{N}^d$ shift) and trees (tree-shift). Firstly, for the completeness of the article, we provide a proof that an $\mathbb{N}^d$ shift has positive (topological) entropy if and only if it has an independence set with positive upper density. Secondly, we obtain that when the base shift $X$ is a hereditary shift, then the associated tree-shift $\mathcal{T}_X$ on an unexpandable tree has positive entropy if and only if it has an independence set with positive density. However, the independence of the tree-shift on an expandable tree differs from that of $\mathbb{N}^d$ shifts or tree-shifts on unexpandable trees. The boundary independence property is introduced and we prove that it is equivalent to the positive entropy of a tree-shift on an expandable tree.

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arXiv2024

Limit sets, internal chain transitivity and orbital shadowing of tree-shifts defined on Markov-Cayley trees

In this paper, we introduce the concepts of $ω$-limit sets and pseudo orbits for a tree-shift defined on a Markov-Cayley tree, extending the results of tree-shifts defined on $d$-trees [5,6]. Firstly, we establish the relationships between $ω$-limit sets and we introduce a modified definition of $ω$-limit set based on complete prefix sets (Theorems 1.4 and 1.9). Secondly, we introduce the concept of projected pseudo orbits and investigate the concept of the shadowing property (Theorems 1.12 and 1.14).

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arXiv2023

The strip entropy approximation of Markov shifts on trees

The strip entropy is studied in this article. We prove that the strip entropy approximation is valid for every ray of a golden-mean tree. This result extends the previous result of [Petersen-Salama, Discrete \& Continuous Dynamical Systems, 2020] on the conventional 2-tree. Lastly, we prove that the strip entropy approximation is valid for eventually periodic rays of a class of Markov-Cayley trees.

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arXiv2023

Large deviation principle of Multiplicative Ising models on Markov-Cayley Trees

In this paper, we study the large deviation principle (LDP) for two types (Type I and Type II) of multiplicative Ising models. For Types I and II, the explicit formulas for the free energy functions and the associated rate functions are derived. Furthermore, we prove that those free energy functions are differentiable, which indicates that both systems are characterized by a lack of phase transition phenomena.

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arXiv2023

The entropy structures of axial products on $\mathbb{N}^d$ and Trees

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 Schraudner [28], we establish the entropy formula and structures for full axial extension shifts on $\mathbb{N}^d$ and $\mathcal{T}_d$. Combining the aforementioned results with the findings on the surface entropy for multiplicative integer systems [8] on $\mathbb{N}^d$ enables us to estimate the surface entropy for the full axial extension shifts on $\mathcal{T}_d$. Finally, we extend the results of full axial extension shifts on $\mathcal{T}_d$ to general trees.

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arXiv2022

Boundary complexity and surface entropy of 2-multiplicative integer systems on $\mathbb{N}^d$

In this article, we introduce the concept of the boundary complexity and prove that for a 2-multiplicative integer system (2-MIS) $X^{p}_Ω$ on $\mathbb{N}$ (or $X^{\bf p}_Ω$ on $\mathbb{N}^d,d\geq 2$), every point in $[h(X^p_Ω), \log r]$ can be realized as a boundary complexity of a 2-MIS with a specific speed, where r stands for the number of the alphabets. The result is new and quite different from $\mathbb{N}^d$ subshifts of finite type (SFT) for $d\geq 1$. Furthermore, the rigorous formula of surface entropy for a $\mathbb{N}^d$ 2-MIS is also presented. This provides an efficient method to calculate the topological entropy for $\mathbb{N}^d$ 2-MIS and also provides an intrinsic differences between $\mathbb{N}^d$ $k$-MIS and SFTs for $d\geq 1$ and $k\geq 2$.

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arXiv2022

Topological Entropy for Shifts of Finite Type Over $\mathbb{Z}$ and Trees

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 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, which makes such a set not dense. Last but not least, the topological entropy of a reducible hom tree-shift of finite type is equal to or larger than that of its maximal irreducible component.

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arXiv2022

Thermodynamic formalism and large deviation principle of multiplicative Ising models

The aim of this study is tree-fold. First, we investigate the thermodynamics of the Ising models with respect to 2-multiple Hamiltonians. This extends the previous results of [Chazotte and Redig, Electron. J. Probably., 2014] to $\mathbb{N}^d$. Second, we establish the large deviation principle (LDP) of the average $\frac{1}{N} S_N^G$, where $S_N^G$ is a 2-multiple sum along a semigroup generated by k numbers which are k co-primes. This extends the previous results [Ban et al. Indag. Math., 2021] to a board class of the long-range interactions. Finally, the results described above are generalized to the multidimensional lattice $\mathbb{N}^d, d\geq1$.

math.PR↗
arXiv2021

Stem and topological entropy on Cayley trees

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.

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arXiv2021

Characterization and Topological Behavior of Homomorphism Tree-Shifts

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 $\mathcal{T}_{X}$, 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 $\mathcal{T}_{X}$. Furthermore, an equivalence between irreducibility on $\mathcal{T}_{A}$ and irreducibility on $X_A$ are seen, and so is one between topologically mixing on $\mathcal{T}_{A}$ 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.

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arXiv2021

Large Deviation Principle of Multidimensional Multiple Averages on $\mathbb{N}^d$

This paper establishs the large deviation principle (LDP) for multiple averages on $\mathbb{N}^d$. We extend the previous work of [Carinci et al., Indag. Math. 2012] to multidimensional lattice $\mathbb{N}^d$ for $d\geq 2$. The same technique is also applicable to the weighted multiple average launched by Fan [Fan, Adv. Math. 2021]. Finally, the boundary conditions are imposed to the multiple sum and explicit formulae of the energy functions with respect to the boundary conditions are obtained.

math.PR↗
arXiv2019

Topologically Mixing Properties of Multiplicative Integer System

Motivated from the study of multiple ergodic average, the investigation of multiplicative shift spaces has drawn much of interest among researchers. This paper focuses on the relation of topologically mixing properties between multiplicative shift spaces and traditional shift spaces. Suppose that $\mathsf{X}_Ω^{(l)}$ is the multiplicative subshift derived from the shift space $Ω$ with given $l > 1$. We show that $\mathsf{X}_Ω^{(l)}$ is (topologically) transitive/mixing if and only if $Ω$ is extensible/mixing. After introducing $l$-directional mixing property, we derive the equivalence between $l$-directional mixing property of $\mathsf{X}_Ω^{(l)}$ and weakly mixing property of $Ω$.

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