arXiv subjects

Chunlin Liu

Publications and source records attributed to Chunlin Liu.

At least 37 records · Page 2Linked to original sources

arXiv2025

Idempotents in the Ellis semigroup of Floyd-Auslander systems

We study minimal idempotents $J^{\mathrm{min}}(X)$ in the Ellis semigroup $E(X)$ associated with a Floyd-Auslander system $(X,T)$. We show that $(X,T)$ is non-tame if and only if $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$, which happens exactly when the factor map onto the maximal equicontinuous factor possesses uncountably many non-invertible fibres. This yields an easy-to-check criterion for distinguishing tame from non-tame Floyd-Auslander systems and, more importantly, provides an entire family of regular almost automorphic systems with $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$. Notably, all previously known regular almost automorphic non-tame systems exhibited only a small (i.e. $\leq 2^{\aleph_0}$) set of minimal idempotents. We obtain our result by leveraging an alternative characterisation of (non)-tameness through, what we call, choice domains.

math.DS↗

arXiv2025

MEBench: Benchmarking Large Language Models for Cross-Document Multi-Entity Question Answering

Multi-entity question answering (MEQA) represents significant challenges for large language models (LLM) and retrieval-augmented generation (RAG) systems, which frequently struggle to consolidate scattered information across diverse documents. While existing methods excel at single-document comprehension, they often struggle with cross-document aggregation, particularly when resolving entity-dense questions like "What is the distribution of ACM Fellows among various fields of study?", which require integrating entity-centric insights from heterogeneous sources (e.g., Wikipedia pages). To address this gap, we introduce MEBench, a novel multi-document, multi-entity benchmark designed to systematically evaluate LLMs' capacity to retrieve, consolidate, and reason over fragmented information. Our benchmark comprises 4,780 questions which are systematically categorized into three primary categories, further divided into eight distinct types, ensuring broad coverage of real-world multi-entity reasoning scenarios. Our experiments on state-of-the-art LLMs (e.g., GPT-4, Llama-3) and RAG pipelines reveal critical limitations: even advanced models achieve only 59% accuracy on MEBench. Our benchmark emphasizes the importance of completeness and factual precision of information extraction in MEQA tasks, using Entity-Attributed F1 (EA-F1) metric for granular evaluation of entity-level correctness and attribution validity. MEBench not only highlights systemic weaknesses in current LLM frameworks but also provides a foundation for advancing robust, entity-aware QA architectures.

cs.CL↗

arXiv2025

A note on multivariate diam mean equicontinuity and frequent stability

Let $(X,G)$ be a topological dynamical system, given by the action of a is a countable discrete infinite group on a compact metric space $X$. We prove that if $(X,G)$ is minimal, then it is either diam-mean $m$-equicontinuious or diam-mean $m$-sensitive. Similarly, $(X,G)$ is either frequently $m$-stable or strongly $m$-spreading. Further, when $G$ is abelian (or, more generally, virtually nilpotent), then the following statements are equivalent: $\bullet$ $(X,G)$ is a regular $m$-to-one extension of its maximal equicontinuous factor; $\bullet$ $(X,G)$ is diam-mean $(m+1)$-equicontinuious, and not diam mean $m$-equicontinuious; $\bullet$ $(X,G)$ is not diam-mean $(m+1)$-sensitive, but diam mean $m$-sensitive; $\bullet$ $(X,G)$ has an essential weakly mean sensitive $m$-tuple but no essential weakly mean sensitive $(m+1)$-tuple. This provides a {\em \enquote*{local}} characterisation of $m$-regularity and mean $m$-sensitivity vial weakly mean sensitive tuples. The same result holds when $G$ is amenable and $(X,G)$ satisfies the local Bronstein condition.

math.DS↗

arXiv2025

The Interplay between Additive and Multiplicative Central Sets Theorems

The concept of Central sets, introduced by Furstenberg through the framework of topological dynamics, has played a pivotal role in combinatorial number theory. Furstenberg's Central Sets Theorem highlighted their rich combinatorial structure. Later, De, Hindman, and Strauss strengthen this theorem using the algebraic framework of the Stone--Čech compactification. In this article, we establish a unified version of the Central Sets Theorem that simultaneously captures both additive and multiplicative structures.

math.CO↗

arXiv2025

Independence and mean sensitivity in minimal systems under group actions

In this paper, we mainly study the relation between regularity, independence and mean sensitivity for minimal systems. In the first part, we show that if a minimal system is incontractible, or local Bronstein with an invariant Borel probability measure, then the regularity is strictly bounded by the infinite independence. In particular, the following two types of minimal systems are applicable to our result: (1) The acting group of the minimal system is a virtually nilpotent group. (2) The minimal system is a proximal extension of its maximal equicontinuous factor and admits an invariant Borel probability measure. Items (1) and (2) correspond to Conjectures 1 and 2 from Huang, Lian, Shao, and Ye (J. Funct. Anal., 2021); item (1) verifies Conjecture 1 in the virtually nilpotent case, and item (2) gives an affirmative answer to Conjecture 2. In the second part, for a minimal system acting by an amenable group, under the local Bronstein condition, we establish parallel results regarding weak mean sensitivity and establish that every mean-sensitive tuple is an IT-tuple.

math.DS↗

arXiv2025

Independence, sequence entropy and mean sensitivity for invariant measures

We investigate the connections between independence, sequence entropy, and mean sensitivity for a measure preserving system under the action of a countable infinite discrete group. We establish that every sequence entropy tuple for an invariant measure is an IT tuple. Furthermore, if the acting group is amenable, we show that for an ergodic measure, the sequence entropy tuples, the mean sensitive tuples along some tempered Følner sequence, and the sensitive in the mean tuples along some tempered Følner sequence coincide.

math.DS↗

arXiv2025

Variational principles for metric mean dimension with potential of level sets

We establish three variational principles for the upper metric mean dimension with potential of level sets of continuous maps in terms of the entropy of partitions and Katok's entropy of the underlying system. Our results hold for dynamical systems exhibiting the specification property. Moreover, we apply our results to study the metric mean dimension of suspension flows.

math.DS↗

arXiv2024

Ergodicity and Mixing of Sublinear Expectation System and Applications

We utilize an ergodic theory framework to explore sublinear expectation theory. Specifically, we investigate the pointwise Birkhoff's ergodic theorem for invariant sublinear expectation systems. By further assuming that these sublinear expectation systems are ergodic, we derive stronger results. Furthermore, we relax the conditions for the law of large numbers and the strong law of large numbers under sublinear expectations from independent and identical distribution to $α$-mixing. These results can be applied to a class of stochastic differential equations driven by $G$-Brownian motion (i.e., $G$-SDEs), such as $G$-Ornstein-Uhlenbeck processes. As byproducts, we also obtain a series of applications for classical ergodic theory and capacity theory.

math.PR↗

arXiv2024

Finite ergodic components for upper probabilities

Under the notion of ergodicity of upper probability in the sense of Feng and Zhao (2021) that any invariant set either has capacity $0$ or its complement has capacity 0, we introduce the definition of finite ergodic components (FEC). We prove an invariant upper probability has FEC if and only if it is in the regime that any invariant set has either capacity $0$ or capacity $1$, proposed by Cerreia-Vioglio, Maccheroni, and Marinacci (2016). Furthermore, this is also equivalent to that the eigenvalue $1$ of the Koopman operator is of finite multiplicity, while in the ergodic upper probability regime, as in the classical ergodic probability case, the eigenvalue $1$ of the Koopman operator is simple. Additionally, we obtain the equivalence of the law of large numbers with multiple values, the asymptotic independence and the FEC. Furthermore, we apply these to obtain the corresponding results for non-invariant probabilities.

math.PR↗

arXiv2024

Ergodicity and Mixing of invariant capacities and applications

We introduce the notion of common conditional expectation to investigate Birkhoff's ergodic theorem and subadditive ergodic theorem for invariant upper probabilities. If in addition, the upper probability is ergodic, we construct an invariant probability to characterize the limit of the ergodic mean. Moreover, this skeleton probability is the unique ergodic probability in the core of the upper probability, that is equal to all probabilities in the core on all invariant sets. We have the following applications of these two theorems: $\bullet$ provide a strong law of large numbers for ergodic stationary sequence on upper probability spaces; $\bullet$ prove the multiplicative ergodic theorem on upper probability spaces; $\bullet$ establish a criterion for the ergodicity of upper probabilities in terms of independence. Furthermore, we introduce and study weak mixing for capacity preserving systems. Using the skeleton idea, we also provide several characterizations of weak mixing for invariant upper probabilities. Finally, we provide examples of ergodic and weakly mixing capacity preserving systems. As applications, we obtain new results in the classical ergodic theory. e.g. in characterizing dynamical properties on measure preserving systems, such as weak mixing, periodicity. Moreover, we use our results in the nonlinear theory to obtain the asymptotic independence, Birkhoff's type ergodic theorem, subadditive ergodic theorem, and multiplicative ergodic theorem for non-invariant probabilities.

math.PR↗

arXiv2024

Directional Pinsker algebra and its applications

In this paper, we introduce the directional Pinsker algebra, and construct a skew product to study it. As applications, we show that 1. if a $\mathbb{Z}^2$-system with positive directional measure-theoretic entropy then it is multivariant directional mean Li-Yorke chaotic along the corresponding direction; 2. for any ergodic measure on a $\mathbb{Z}^2$-system, the intersection of the set of directional measure-theoretic entropy tuples with the set of directional asymptotic tuples is dense in the set of directional measure-theoretic entropy tuples.

math.DS↗

arXiv2024

Ergodic measures in minimal group actions with finite topological sequence entropy

Let $G$ be an infinite discrete countable group and $(X,G)$ be a minimal $G$-system. In this paper, we prove the supremum of topological sequence entropy of $(X,G)$ is not less than $\log(\sum_{μ\in\mathcal{M}^e(X,G)}e^{h_μ^*(X,G)})$. If additionally $G$ is abelian then there is a constant $K\in\mathbb{N}\cup\{\infty\}$ with $\log K\le h_{top}^*(X,G)$ such that $ν(\{y\in H:|π^{-1}(y)|=K\})=1$ where $(H,G)$ is the maximal equicontinuous factor of $(X,G)$, $π:(X,G)\to (H,G)$ is the factor map and $ν$ is the Haar measure of $H$.

math.DS↗

arXiv2023

The irregular set for maps with almost weak specification property has full metric mean dimension

Let $(X, d)$ be a compact metric space, $f: X \to X$ be a continuous transformation with the almost weak specification property and $φ: X \to \mathbb{R}$ be a continuous function. We consider the set (called the irregular set for $φ$) of points for which the Birkhoff average of $φ$ does not exist and show that this set is either empty or carries full Bowen upper and lower metric mean dimension.

math.DS↗

arXiv2023

Directional weak mixing for $\mathbb{Z}^q$-actions

In this paper, we introduce and characterize the concept of directional weak mixing through independence, sequence entropy, the mean ergodic theorem, and other notions. Additionally, we deduce a directional version of the Koopman-von Neumann spectrum mixing theorem. Furthermore, we explore the relation between directional weak mixing and weak mixing.

math.DS↗

arXiv2023

Sequence entropy for amenable group actions

We study the sequence entropy for amenable group actions and investigate systematically spectrum and several mixing concepts via sequence entropy both in measure-theoretic dynamical systems and topological dynamical systems. Moreover, we use sequence entropy pairs to characterize weakly mixing and null systems.

math.DS↗

arXiv2023

Pinsker $σ$-algebra Character and mean Li-Yorke chaos

Let $G$ be an infinite countable discrete amenable group. For any $G$-action on a compact metric space $X$, it is proved that for any sequence $(G_n)_{n\ge 1}$ consisting of non-empty finite subsets of $G$ with $\lim_{n\to \infty}|G_n|=\infty$, Pinsker $σ$-algebra is a characteristic factor for $(G_n)_{n\ge 1}$. As a consequence, for a class of $G$-topological dynamical systems, positive topological entropy implies mean Li-Yorke chaos along a class of sequences consisting of non-empty finite subsets of $G$.

math.DS↗

arXiv2023

Sequence entropy tuples and mean sensitive tuples

Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the $μ$-sequence entropy tuple, the $μ$-mean sensitive tuple and the $μ$-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.

math.DS↗

arXiv2022

Mean Li-Yorke chaos along any infinite sequence for infinite-dimensional random dynamical systems

In this paper, we study the mean Li-Yorke chaotic phenomenon along any infinite positive integer sequence for infinite-dimensional random dynamical systems. To be precise, we prove that if an injective continuous infinite-dimensional random dynamical system $(X,ϕ)$ over an invertible ergodic Polish system $(Ω,\mathcal{F},\mathbb{P},θ)$ admits a $ϕ$-invariant random compact subset $K$ with $h_{top}(K,ϕ)>0$, then given a positive integer sequence $\mathbf{a}=\{a_i\}_{i\in\mathbb{N}}$ with $\lim_{i\to+\infty}a_i=+\infty$, for $\mathbb{P}$-a.s. $ω\inΩ$ there exists an uncountable subset $S(ω)\subset K(ω)$ and $ε(ω)>0$ such that for any distinct points $x_1$, $x_2\in S(ω)$ with following properties \begin{align*} \liminf_{N\to+\infty}\frac{1}{N}\sum_{i=1}^{N} d\big(ϕ(a_i, ω)x_1, ϕ(a_i, ω)x_2\big)=0,\quad\limsup_{N\to+\infty}\frac{1}{N}\sum_{i=1}^{N} d\big(ϕ(a_i, ω)x_1, ϕ(a_i, ω)x_2\big)>ε(ω), \end{align*} where $d$ is a compatible complete metric on $X$.

math.DS↗