arXiv subjects

Zhengyu Yin

Publications and source records attributed to Zhengyu Yin.

10 recordsLinked to original sources

arXiv2026

One residual set of Cantor graphs with dense conjugacy classes

In this paper, we consider the space $\mathcal{G}raph(\mathcal{C})$ of all closed graphs on the Cantor set $\mathcal{C}$. Equipping this space with Hausdorff metric and employing combinatorial methods, we see that the residual isomorphism classes in the space of single-valued continuous maps on the Cantor set are no longer a $G_δ$ set but still dense in $\mathcal{G}raph(\mathcal{C})$, and we prove that the set consisting of elements with dense conjugacy classes is a dense $G_δ$ subset of $\mathcal{G}raph(\mathcal{C})$. Finally, we prove that the set consisting of elements with zero topological entropy is a dense $G_δ$ set in $\mathcal{G}raph(\mathcal{C}).$

math.DS↗

arXiv2026

Finite type as fundamental objects even non-single-valued and non-continuous

In this paper, inspired by the elegant work of Good and Meddaugh \cite{GM} and the graph models for zero-dimensional systems developed by several authors, like Gambaudo and Martens \cite{GM06}, Shimomura \cite{Sh14}. We try to discover a connection among some objects, such as finite directed graph, shift of finite type and shadowing property by employing the Closed Graph Theorem for multivalued maps. From the perspective of structure theorems, we demonstrate that every closed relation (multivalued map) on a compact, totally disconnected space is represented as an inverse limit of finite directed graph homomorphisms satisfying the Mittag-Leffler condition. Moreover, from dichotomy-theorem point of view, we prove that an inverse limit of finite directed graph homomorphisms possesses the shadowing property if and only if its induced space of infinite graph walks (as a shift of finite type) satisfies the Mittag-Leffler condition. As an application, a question raised by Boroński, Bruin and Kucharski \cite{BBK} is also concerned. Furthermore, we show that under a multivalued dynamical system, the resulting dynamical behaviors exhibit greater diversity and counterintuitively compared to those observed in single-valued continuous systems.

math.DS↗

arXiv2025

Variational principles of relative weighted topological pressures

Recently, M. Tsukamoto (New approach to weighted topological entropy and pressure, Ergod. Theory Dyn. Syst. 43 (2023) 1004-1034) used a new approach to define the weighted topological entropy and pressure. Inspired by his ideas, we introduce the relative weighted topological entropy and pressure for factor maps and establish several variational principles. One of these results involves a question raised by D. Feng and W. Huang (Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016) 411-452), whether there is a relative version of the weighted variation principle. In this paper, we try to establish such variational principle. Furthermore, we generalize the Ledrappier and Walters' type relative variational principle to the weighted version.

math.DS↗

arXiv2025

Shadowing property for set-valued map and its inverse limit

In this article, we investigate the relationship between the shadowing property of set-valued maps and their associated inverse limit systems. We show that if a set-valued map is expansive and open in the context of set-valued dynamics, then certain induced inverse limit systems have the shadowing property. Additionally, we prove that a continuous set-valued map has the shadowing property if and only if some of its induced inverse limit system also has shadowing property. Finally, we establish that the shadowing property of a set-valued map is equivalent to the shadowing property of its induced inverse set-valued system.

math.DS↗

arXiv2024

Scaled packing pressures on subsets for amenable group actions

In this paper, we study the properties of the scaled packing topological pressures for topological dynamical system $(X,G)$, where $G$ is a countable discrete infinite amenable group. We show that the scaled packing topological pressures can be determined by the scaled Bowen topological pressures. We obtain Billingsley's Theorem for the scaled packing pressures with a $G$-action. Then we get a variational principle between the scaled packing pressures and the scaled measure-theoretic upper local pressures. Finally, we give some restrictions on the scaled sequence $\mathbf{b}$, then in the case of the set $X_μ$ of generic points, we prove that $$P^{P}(X_μ,\left\{F_{n}\right\},f,\mathbf{b})=h_μ(X)+\int_{X} f \mathrm{d}μ,$$ if $\left\{F_{n}\right\}$ is tempered and $μ$ is a $G$-invariant ergodic Borel probability measure.

math.DS↗

arXiv2024

Minimal amenable subshift with full mean dimension

Let $G$ be an infinite countable amenable group and $P$ a polyhedron with topological dimension $dim(P)<\infty$. We construct a minimal subshift $(X,G)$ such that its mean topological dimension is equal to $dim(P)$. This result answers the question of D. Dou in \cite{DD}, moreover, it is also an extension of the work of L. Jin and Y. Qiao \cite{JQ} for $\mathbb{Z}$-action.

math.DS↗

arXiv2023

Variational principle of higher dimension weighted pressure for amenable group actions

Let $r\geq 2$ and $(X_i,G)$ $(i=1,\cdots,r)$ be topological dynamical systems with $G$ being an infinite discrete amenable group. Suppose that $π_i:(X_i,G)\to (X_{i+1},G)$ are factor maps and $0\leq w_i\leq 1$. In this article, for $f\in C(X_1)$, we introduce the weighted topological pressure $P^{\textbf{a}}(f,G)$ for higher dimensions (not only for $r=2$) of amenable group actions. By using measure-theoretical theory, we establish a variational principle as \begin{align*} P^{\textbf{a}}(f,G)=\sup_{μ\in \mathcal{M}^G(X_1)}\Big(\sum_{i=1}^rw_ih_{μ_i}(X_i,G)+w_1\int_{X_1}fdμ\Big), \end{align*} where $μ_i=π_{i-1}\circ\cdots\circπ_{1}μ$ is the induced $G$-invariant measure on $X_{i}$.

math.DS↗

arXiv2023

Veech's Theorem of $G$ acting freely on $G^{\textrm{LUC}}$ and Structure Theorem of a.a. flows

Veech's Theorem claims that if $G$ is a locally compact\,(LC) Hausdorff topological group, then it may act freely on $G^{\textrm{LUC}}$. We prove Veech's Theorem for $G$ being only locally quasi-totally bounded, not necessarily LC. And we show that the universal a.a. flow is the maximal almost 1-1 extension of the universal minimal a.p. flow and is unique up to almost 1-1 extensions. In particular, every endomorphism of Veech's hull flow induced by an a.a. function is almost 1-1; for $G=\mathbb{Z}$ or $\mathbb{R}$, $G$ acts freely on its canonical universal a.a. space. Finally, we characterize Bochner a.a. functions on a LC group $G$ in terms of Bohr a.a. function on $G$ (due to Veech 1965 for the special case that $G$ is abelian, LC, $σ$-compact, and first countable).

math.DS↗

arXiv2022

Relative entropy dimensions for amenable group actions

We study the topological complexities of relative entropy zero extensions acted by countableinfinite amenable groups. Firstly, for a given Folner sequence $\{F_n\}_{n=0}^\infty$, we define respectively the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. Meanwhile, we investigate the relations among them. Secondly, we introduce the notion of a relative dimension set. Moreover, using it, we discuss the disjointness between the relative entropy zero extensions which generalizes the results of Dou, Huang and Park[Trans. Amer. Math. Soc. 363(2) (2011), 659-680].

math.DS↗

arXiv2014

Stackelberg vs. Nash in Security Games: An Extended Investigation of Interchangeability, Equivalence, and Uniqueness

There has been significant recent interest in game-theoretic approaches to security, with much of the recent research focused on utilizing the leader-follower Stackelberg game model. Among the major applications are the ARMOR program deployed at LAX Airport and the IRIS program in use by the US Federal Air Marshals (FAMS). The foundational assumption for using Stackelberg games is that security forces (leaders), acting first, commit to a randomized strategy; while their adversaries (followers) choose their best response after surveillance of this randomized strategy. Yet, in many situations, a leader may face uncertainty about the follower's surveillance capability. Previous work fails to address how a leader should compute her strategy given such uncertainty. We provide five contributions in the context of a general class of security games. First, we show that the Nash equilibria in security games are interchangeable, thus alleviating the equilibrium selection problem. Second, under a natural restriction on security games, any Stackelberg strategy is also a Nash equilibrium strategy; and furthermore, the solution is unique in a class of security games of which ARMOR is a key exemplar. Third, when faced with a follower that can attack multiple targets, many of these properties no longer hold. Fourth, we show experimentally that in most (but not all) games where the restriction does not hold, the Stackelberg strategy is still a Nash equilibrium strategy, but this is no longer true when the attacker can attack multiple targets. Finally, as a possible direction for future research, we propose an extensive-form game model that makes the defender's uncertainty about the attacker's ability to observe explicit.

cs.GT↗