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Yu-Liang Wu

Publications and source records attributed to Yu-Liang Wu.

12 recordsLinked to original sources
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.

math.DS↗
arXiv2024

Hausdorff dimensions of irreducible Markov hom tree-shifts

This paper features a Cram\'er'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.

math.PR↗
arXiv2024

Van der Corput and metric theorems for geometric progressions for self-similar measures

We prove a van der Corput lemma for non-atomic self-similar measures $μ$. As an application, we show that the correlations of all finite orders of $( x^n \mod 1 )_{n\geq 1}$ converge to the Poissonian model for $μ$-a.e. $x$, assuming $x>1$. We also complete a recent result of Algom, Rodriguez Hertz, and Wang (obtained simultaneously by Baker and Banaji), showing that any self-conformal measure with respect to a non-affine real analytic IFS has polynomial Fourier decay.

math.DS↗
arXiv2023

Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets

Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the $s$-Beurling densities of regular maximal orthogonal sets of a class of self-similar spectral measures, where $s$ is the Hausdorff dimension of its support and obtain their exact upper bound of the densities.

math.FA↗
arXiv2023

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.

math.DS↗
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.

math.DS↗
arXiv2022

Spectral measures with arbitrary dimensions

It is known [Dai and Sun, J. Funct. Anal. 268 (2015), 2464--2477] that there exist spectral measures with arbitrary Hausdorff dimensions, and it is natural to pose the question of whether similar phenomena occur for other dimensions of spectral measures. In this paper, we first obtain the formulae of Assouad dimension and of lower dimension for a class of Moran measures in dimension one that is introduced by An and He [J. Funct. Anal. 266 (2014), 343--354]. Based on these results, we show the existence of spectral measures with arbitrary Assound dimensions $\dim_A$ and lower dimensions $\dim_L$ ranging from $0$ to $1$, including non-atomic zero-dimensional spectral measures and one-dimensional singular spectral measures, and prove that the two values may coincide. In fact, more is obtained that for any $0 \leq t \leq s \leq r \leq u\leq 1$, there exists a spectral measure $μ$ such that \[\dim_L μ=t, \dim_H μ=s, \dim_Pμ=r~ \text{and} \dim_Aμ=u,\] where $\dim_H$ and $\dim_P$ denote the Hausdorff dimension and packing dimension of the measure $μ$, respectively. This result improves and generalizes the result of Dai and Sun more simply and flexibly.

math.FA↗
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.

math.DS↗
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.

math.DS↗
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 $Ω$.

math.DS↗
arXiv2019

Decidability of irreducible tree shifts of finite type

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.

math.DS↗