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A general framework for studying McKean-Vlasov SDEs via monotone dynamical systems is established in this paper. Under a cooperative condition, we show McKean-Vlasov SDEs admit a comparison principle with respect to the stochastic order, and generate monotone dynamical systems on the $2$-Wasserstein space. Our main results prove the existence of unstable invariant measures, total orderedness of invariant measures, and the existence of monotone connecting orbits between order-related invariant measures for general cooperative McKean-Vlasov SDEs. To achieve our goals, we adopt the theory of monotone dynamical systems, extend the connecting orbit theorem, and deduce a dichotomy structure of equilibria. This method is different from existing approaches, like propagation of chaos and Fokker-Planck equations. A wide range of classical examples are covered by our framework, such as granular media equations in double-well and multi-well confinement potentials with quadratic interaction, double-well landscapes with perturbation, and higher dimensional equations, even driven by multiplicative noises., Comment: 56 pages, 2 figures

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.

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 $\log(\sum_{\mu\in\mathcal{M}^e(X,G)}e^{h_\mu^*(X,G)})$. 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 $\nu(\{y\in H:|\pi^{-1}(y)|=K\})=1$ where $(H,G)$ is the maximal equicontinuous factor of $(X,G)$, $\pi:(X,G)\to (H,G)$ is the factor map and $\nu$ is the Haar measure of $H$., Comment: arXiv admin note: text overlap with arXiv:2002.08792 by other authors

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 $\mu$-sequence entropy tuple, the $\mu$-mean sensitive tuple and the $\mu$-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.

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

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.

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 $(X,\phi)$ over an invertible ergodic Polish system $(\Omega,\mathcal{F},\mathbb{P},\theta)$ admits a $\phi$-invariant random compact subset $K$ with $h_{top}(K,\phi)>0$, 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. $\omega\in\Omega$ there exists an uncountable subset $S(\omega)\subset K(\omega)$ and $\epsilon(\omega)>0$ such that for any distinct points $x_1$, $x_2\in S(\omega)$ with following properties \begin{align*} \liminf_{N\to+\infty}\frac{1}{N}\sum_{i=1}^{N} d\big(\phi(a_i, \omega)x_1, \phi(a_i, \omega)x_2\big)=0,\quad\limsup_{N\to+\infty}\frac{1}{N}\sum_{i=1}^{N} d\big(\phi(a_i, \omega)x_1, \phi(a_i, \omega)x_2\big)>\epsilon(\omega), \end{align*} where $d$ is a compatible complete metric on $X$., Comment: 22 pages

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.

In this short paper, we compute the directional sequence entropy for of $\mathbb{Z}_+\times \mathbb{Z}$-actions generated by cellular automata and the shift map. Meanwhile, we study the directional dynamics of this system. As a corollary, we prove that there exists a sequence such that for any direction, some of the systems above have positive directional sequence entropy. Moreover, with help of mean ergodic theory for directional weak mixing systems, we obtain a result of number theory about combinatorial numbers.

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 $\sigma$-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 $G$.

The deployment of various networks (e.g., Internet of Things [IoT] and mobile networks), databases (e.g., nutrition tables and food compositional databases), and social media (e.g., Instagram and Twitter) generates huge amounts of food data, which present researchers with an unprecedented opportunity to study various problems and applications in food science and industry via data-driven computational methods. However, these multi-source heterogeneous food data appear as information silos, leading to difficulty in fully exploiting these food data. The knowledge graph provides a unified and standardized conceptual terminology in a structured form, and thus can effectively organize these food data to benefit various applications. In this review, we provide a brief introduction to knowledge graphs and the evolution of food knowledge organization mainly from food ontology to food knowledge graphs. We then summarize seven representative applications of food knowledge graphs, such as new recipe development, diet-disease correlation discovery, and personalized dietary recommendation. We also discuss future directions in this field, such as multimodal food knowledge graph construction and food knowledge graphs for human health., Comment: 45 pages, 6 figures

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.

Given $q\in\mathbb{N}$, let $(X,T)$ be a $\mathbb{Z}^q$-system, $\vec{v}\in\mathbb{R}^q\setminus\{\vec{0}\}$ be a direction vector and $\textbf{b}\in\mathbb{R}_+^{q-1}$. We study $(X,T)$ that has bounded complexity with respect to three kinds of metrics defined along direction $\vec{v}$: the directional Bowen metric $d_k^{\vec{v},\textbf{b}}$, the directional max-mean metric $\hat{d}_k^{\vec{v},\textbf{b}}$ and the directional mean metric $\bar{d}_k^{\vec{v},\textbf{b}}$. It is shown that $(X,T)$ has bounded topological complexity with respect to $\{d_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ (resp. $\{\hat{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$) if and only if $T$ is $(\vec{v},\textbf{b})$-equicontinuous (resp. $(\vec{v},\textbf{b})$-equicontinuous in the mean). Meanwhile, it turns out that an invariant Borel probability measure $\mu$ on $X$ has bounded complexity with respect to $\{d_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if $T$ is $(\mu,\vec{v},\textbf{b})$-equicontinuous. Moreover, it is shown that $\mu$ has bounded complexity with respect to $\{\bar{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if $\mu$ has bounded complexity with respect to $\{\hat{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if $T$ is $(\mu,\vec{v},\textbf{b})$-mean equicontinuous if and only if $T$ is $(\mu,\vec{v},\textbf{b})$-equicontinuous in the mean if and only if $\mu$ has $\vec{v}$-discrete spectrum.

In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb{Z}^q$-systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, direcitonal discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb{Z}^q$-system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb{Z}^q$-system has directional discrete spectrum along $q$ linearly independent directions if and only if it has discrete spectrum.