In this paper, we reduce the logarithmic Sarnak conjecture to the $\{0,1\}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.

We show that any action of a countable amenable group on a uniquely arcwise connected continuum has a periodic point of order $\leq 2$ .

In recent years, there have been a significant influenza activity and emerging influenza strains in China, resulting in an increasing number of influenza virus infections and leading to public health concerns. The aims of this study were to identify the epidemiological and aetiological characteristics of influenza and establish seasonal autoregressive integrated moving average (SARIMA) models for forecasting the percentage of visits for influenza-like illness (ILI%) in urban and rural areas of Shenyang. Influenza surveillance data were obtained for ILI cases and influenza virus positivity from 18 sentinel hospitals. The SARIMA models were constructed to predict ILI% for January–December 2019. During 2010–2018, the influenza activity was higher in urban than in rural areas. The age distribution of ILI cases showed the highest rate in young children aged 0–4 years. Seasonal A/H3N2, influenza B virus and pandemic A/H1N1 continuously co-circulated in winter and spring seasons. In addition, the SARIMA (0, 1, 0) (0, 1, 2)12 model for the urban area and the SARIMA (1, 1, 1) (1, 1, 0)12 model for the rural area were appropriate for predicting influenza incidence. Our findings suggested that there were regional and seasonal distinctions of ILI activity in Shenyang. A co-epidemic pattern of influenza strains was evident in terms of seasonal influenza activity. Young children were more susceptible to influenza virus infection than adults. These results provide a reference for future influenza prevention and control strategies in the study area.

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$, the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$, if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-mean equicontinuous and if and only if it has discrete spectrum.

In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any non-empty open subset $U\subset X$, there is $x\in D\cap U$ such that $\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^{n},T^{(n)})$ and $(X,T^{n})$ for any $n\in \mathbb{N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_{K})$ is disjoint from all minimal systems.

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between an invertible topological dynamical system (t.d.s.) $(X,T)$ and the t.d.s. $(K(X),T_{K})$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a weakly mixing t.d.s. which induces dense periodic points on the hyperspace $K(X)$ but itself does not have dense distal points, solving in that way a few open questions from earlier articles by Dong, and Li, Yan and Ye.

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.

An ∞-step nilsystem is an inverse limit of minimal nilsystems. In this article, it is shown that a minimal distal system is an ∞-step nilsystem if and only if it has no non-trivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without non-trivial pairs with arbitrarily long finite IP-independence sets is an almost one-to-one extension of its maximal ∞-step nilfactor, and each invariant ergodic measure is isomorphic (in the measurable sense) to the Haar measure on some ∞-step nilsystem. The question if such a system is uniquely ergodic remains open. In addition, the topological complexity of an ∞-step nilsystem is computed, showing that it is polynomial for each non-trivial open cover.

Let (X,T) be a topological dynamical system (TDS), and h(T,K) the topological entropy of a subset K of X. (X,T) is lowerable if for each 0≤h≤h(T,X) there is a non-empty compact subset with entropy h; it is hereditarily lowerable if each non-empty compact subset is lowerable; it is hereditarily uniformly lowerable if for each non-empty compact subset K and each 0≤h≤h(T,K) there is a non-empty compact subset Kh⊆K with h(T,Kh)=h and Kh has at most one limit point. It is shown that each TDS with finite entropy is lowerable, and that a TDS (X,T) is hereditarily uniformly lowerable if and only if it is asymptotically h-expansive.

In this survey we offer an overview of the so-called local entropy theory, which has been in development since the early 1990s. While doing so, we emphasize the connections between the topological dynamics and the ergodic theory points of view.

For a given factor map $\pi:X\longrightarrow Y$ between two topological dynamical systems and a Borel cover ${\mathcal U}$, two notions of measure-theoretical conditional entropy $h_\mu^+(T,{\mathcal U}\mid Y)$ and $h_\mu^-(T,{\mathcal U}\mid Y)$ for an invariant Borel probability measure $\mu$ are introduced. It is shown that $h_\mu^+(T,{\mathcal U}\mid Y)=h_\mu^-(T,{\mathcal U}\mid Y)$. Moreover, $\max_{\mu}h_\mu^+(T,{\mathcal U}\mid Y)=h_{{\rm top}}(T,{\mathcal U}\mid Y)$ when $\mathcal U$ is an open cover. The relative variational principle is a consequence of the results.

In Blanchard et al (Topological complexity. Ergod. Th. & Dynam. Sys.20 (2000), 641–662), the authors introduced the notion of scattering and a weaker notion of 2-scattering. It is an open question whether the two notions are equivalent. The question is answered affirmatively in this paper. Using the complexity function of an open cover along some sequences of natural numbers, we characterize mild mixing, strong scattering and scattering. We show that mildly mixing (respectively strongly mixing) systems are disjoint from minimal uniformly rigid (respectively minimal rigid) systems.

Moreover, assuming minimality we show that a dynamical system is full-scattering (respectively mildly mixing or weakly mixing) if and only if it is strongly mixing (respectively IP*-transitive or $\mathcal{D}$-transitive), where $\mathcal{D}$ is the collection of subsets of $\mathbb{Z}_+$ with the lower Banach density 1.

We show that a continuous map of a metric space is pointwise recurrent and has the pseudo orbit tracing property if and only if the map is uniformly conjugate to an adding-machine-like map restricted to some invariant subset.

A homeomorphism on a metric space (X,d) is completely scrambled if for each x\not= y\in X, \lim sup_{n\longrightarrow +\infty} d(f^n(x),f^n(y))>0 and \lim inf_{n\ longrightarrow +\infty}d(f^n(x),f^n(y))=0. We study the basic properties of completely scrambled homeomorphisms on compacta and show that there are ‘many’ compacta admitting completely scrambled homeomorphisms, which include some countable compacta (we give a characterization), the Cantor set and continua of arbitrary dimension.

Lef: G → G be a continuous map of a graph and let d(A) denote the derived set (or limit points) of A ⊂ G. We prove that d(Ω(f)) ⊂ λ (f) and the depth of f is at most three. We also prove that if f is piecewise monotone or has zero topological entropy, then the depth of f is at most two. Furthermore, we obtain some results on the topological structure of Ω(f).

Let $T$ be a tree, ${\mathop{\rm End}\nolimits}(T)$ be the number of ends of $T$ and let $L(T)$ be the infimum of topological entropies of transitive maps of $T$. We give an elementary approach to the estimate that $L(T)\ge (1/{\mathop{\rm End}\nolimits} (T))\log 2$. We also divide the set of all trees (up to homeomorphisms) into pairwise disjoint subsets ${\cal P}(i)$, $i\in \{0\}\cup\N$ and prove that $L(T)=(1/({\mathop{\rm End}\nolimits} (T)-i))\log 2$ if $T\in {\cal P}(i)$ with $i=0,1$, and $L(T)\le (\text{respectively} =) (1/({\mathop{\rm End}\nolimits} (T)-i))\log 2$ if $T\in {\cal P}(i)$ (respectively $T\in {\cal P}'(i))$ with $i\ge 2$, where ${\cal P}'(i)$ is an infinite subset of ${\cal P}(i)$. Furthermore, we show that there is a tree $T$ such that the topological entropy of each transitive map of $T$ is larger than $L(T)$, and hence disprove a conjecture of Alseda et al (1997).

We generalise a result of Hosaka and Kato by proving that if the set of periodic points of a continuous map of a tree is closed then each chain recurrent point is a periodic one. We also show that the topological entropy of a tree map is zero if and only if the w-limit set of each chain recurrent point (which is not periodic) contains no periodic points.