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Xiongping Dai

Publications and source records attributed to Xiongping Dai.

At least 19 recordsLinked to original sources
arXiv2026

On generalized Namioka spaces and joint continuity of functions on product of spaces

A space $X$ is called a generalized Namioka space (g$\mathcal{N}$-space), if for every compact space $Y$ and every separately continuous function $f\colon X\times Y\rightarrow\mathbb{R}$, there exists at least one point $x\in X$ such that $f$ is jointly continuous at each point of $\{x\}\times Y$. We principally prove the following results: (1) If $X=\prod_{\alpha\in A}X_\alpha$ is non-meager such that each factor is a separable space or each factor is a pseudo-metric space, then $X$ is a g$\mathcal{N}$-space. (2) If $X$ is a separable space and $Y$ a pseudo-metric space such that $X\times Y$ is Baire (resp. non-meager), then $X\times Y$ is an $\mathcal{N}$-space (resp. a g$\mathcal{N}$-space). (3) If $X=\prod_{\alpha\in A}X_\alpha$ such that each factor is separable and $\prod_{\alpha\in A^\prime}X_\alpha$ is a non-meager space for each countable subset $A^\prime$ of $A$, then $X$ is a non-meager g$\mathcal{N}$-space. (4) If $X=\prod_{\alpha\in A}X_\alpha$ such that each factor has a countable $\pi$-base, then each tail set having the property of Baire in $X$ is either meager or residual. If $G$ is a g$\mathcal{N}$ right-topological group and $X$ a locally compact regular space, or, if $G$ is a separable first countable non-meager right-topological group and $X\times X$ a countably compact completely regular space, then any separately continuous action $G\curvearrowright X$ is jointly continuous.

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arXiv2026

On semi-openness of fiber-onto extensions of minimal semiflows and quasi-separable maps

The purpose of this paper is to find conditions for a continuous onto map $\phi\colon X\rightarrow Y$ and its induced map $\phi_*\colon\mathcal{M}^1(X)\rightarrow\mathcal{M}^1(Y)$ to be semi-open, where $X$, $Y$ are compact Hausdorff spaces and $\mathcal{M}^1(X)$, $\mathcal{M}^1(Y)$ are their Borel probability spaces. For that, we mainly prove the following results by using the structure theory of extensions of semiflows and inverse limit techniques: (1) If $\phi$ is an extension of minimal flows, then $\phi_*$ is semi-open. (2) If $\phi$ is a quasi-separable fiber-onto extension of minimal semiflows, then $\phi$ and $\phi_*$ are semi-open. (3) If $Y$ is metrizable, then $\phi$ is semi-open if and only if $\phi_*$ is semi-open. In addition, if $X,Y$ are left-topological groups, $X$ is Lindel\"{o}f quasi-regular, $Y$ is Baire and if $\phi$ is a locally closed continuous onto equivariant mapping, then $\phi$ is semi-open (This is a generalization of Pontryagin's open-mapping theorem).

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arXiv2024

Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps

Let $f\colon X\rightarrow Y$ be a continuous surjection of compact Hausdorff spaces. By $$f_*\colon\mathfrak{M}(X)\rightarrow\mathfrak{M}(Y),\ μ\mapsto μ\circ f^{-1} \quad{\rm and}\quad 2^f\colon2^X\rightarrow2^Y,\ A\mapsto f[A]$$ we denote the induced continuous surjections on the probability measure spaces and hyperspaces, respectively. In this paper we mainly show the following facts: (1) If $f_*$ is semi-open, then $f$ is semi-open. (2) If $f$ is semi-open densely open, then $f_*$ is semi-open densely open. (3) $f$ is open iff $2^f$ is open. (4) $f$ is semi-open iff $2^f$ is semi-open. (5) $f$ is irreducible iff $2^f$ is irreducible.

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arXiv2024

On sncc-inheritance of pointwise almost periodicity in flows

Let $H$ be a subnormal co-compact closed subgroup of a Hausdorff topological group $T$ and $X$ a compact Hausdorff space. We prove the inheritance theorem: A point of $X$ is almost periodic (a.p.) for $T\curvearrowright X$ iff it is a.p. for $H\curvearrowright X$. Moreover, if $T\curvearrowright X$ is minimal with $H\lhd T$, then $\mathscr{O}_H\colon X\rightarrow2^X$, ${x\mapsto\overline{Hx}}$ is a continuous mapping, and, $T\curvearrowright X/H$ is an a.p. nontrivial factor of $T\curvearrowright X$ iff $T\curvearrowright X\times T/H$ is not minimal.

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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).

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arXiv2023

On M-dynamics and Li-Yorke chaos of extensions of minimal dynamics

Let $π\colon\mathscr{X}\rightarrow\mathscr{Y}$ be an extension of minimal compact metric flows such that $\texttt{R}_π\not=Δ_X$. A subflow of $\texttt{R}_π$ is called an M-flow if it is T.T. and contains a dense set of a.p. points. In this paper we mainly prove the following: (1) $π$ is PI iff $Δ_X$ is the unique M-flow containing $Δ_X$ in $\texttt{R}_π$. (2) If $π$ is not PI, then there exists a canonical Li-Yorke chaotic M-flow in $\texttt{R}_π$. In particular, an Ellis weak-mixing non-proximal extension is non-PI and so Li-Yorke chaotic. (3) A unbounded or non-minimal M-flow, not necessarily compact, is sensitive on initial conditions. (4) every syndetically distal flow is pointwise Bohr a.p.

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arXiv2022

On recurrence in zero-dimsnional locally compact flow with compactly generated phase group

We define recurrence for a compactly generated para-topological group $G$ acting continuously on a locally compact Hausdorff space $X$ with $\dim X=0$, and then, show that if $\overline{Gx}$ is compact for all $x\in X$, the conditions (i) this dynamics is pointwise recurrent, (ii) $X$ is a union of $G$-minimal sets, (iii) the $G$-orbit closure relation is closed in $X\times X$, and (iv) $X\ni x\mapsto \overline{Gx}\in 2^X$ is continuous, are pairwise equivalent. Consequently, if this dynamics is pointwise product recurrent, then it is pointwise regularly almost periodic and equicontinuous; moreover, a distal, compact, and non-connected $G$-flow has a non-trivial equicontinuous pointwise regularly almost periodic factor.

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arXiv2022

On recurrence in zero-dimensional locally compact flow with compactly generated phase group

Let $X$ be a zero-dimensional locally compact Hausdorff space not necessarily metric and $G$ a compactly generated topological group not necessarily abelian or countable. We define recurrence at a point for any continuous action of $G$ on $X$, and then, show that if $\overline{Gx}$ is compact for all $x\in X$, the conditions (i) this dynamics is pointwise recurrent, (ii) $X$ is a union of $G$-minimal sets, (iii) the $G$-orbit closure relation is closed in $X\times X$, and (iv) $X\ni x\mapsto \overline{Gx}\in 2^X$ is continuous, are pairwise equivalent. Consequently, if this dynamics is distal, then it is equicontinuous.

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arXiv2019

The McMahon pseudo-metrics of minimal semiflows with invariant measures

Using McMahon pseudo-metrics, for any minimal semiflow admitting an invariant measure, we study the relationships between its equicontinuous structure relation, regionally proximal relation and Veech's relations; and characterize its weak-mixing. We show its Veech Structure Theorem if it is almost automorphic.

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arXiv2019

Almost automorphy of surjective semiflows on compact Hausdorff spaces

Let $(T,X)$ with phase mapping $(t,x)\mapsto tx$ be a semiflow on a compact $\textrm{T}_2$-space $X$ with phase semigroup $T$ such that $tX=X$ for each $t$ of $T$. An $x\in X$ is called an \textit{a.a. point} if $t_nx\to y, x_n^\prime\to x^\prime$ and $t_nx_n^\prime=y$ implies $x=x^\prime$ for every net $\{t_n\}$ in $T$. In this paper, we study the a.a. dynamics of $(T,X)$; and moreover, we present a complete proof of Veech's structure theorem for a.a. flows.

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arXiv2018

Grünwald version of van der Waerden's theorem for semi-modules

Let $(G,\pmb{+})$ be any given semimodule over a discrete semiring $(R,+,\cdot)$ with a finite coloring, say $G=B_1\cup\dotsm\cup B_q$. By establishing a Regional Multiple Recurrence Theorem for semimodules, we prove that one of the colors $j$ has the property that if $F\subseteq G$ is any finite set, then one can find some "syndetic" subset $D_F$ of $(R,+)$ such that for each $d\in D_F$ there is some $a\in B_j$ with $a\pmb{+}dF\subseteq B_j$. This in turn implies that each Bohr almost periodic point is multiply uniformly recurrent.

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arXiv2018

Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Let $π\colon T\times X\rightarrow X$ with phase map $(t,x)\mapsto tx$, denoted $(π,T,X)$, be a \textit{semiflow} on a compact Hausdorff space $X$ with phase semigroup $T$. If each $t\in T$ is onto, $(π,T,X)$ is called surjective; and if each $t\in T$ is 1-1 onto $(π,T,X)$ is called invertible and in latter case it induces $π^{-1}\colon X\times T\rightarrow X$ by $(x,t)\mapsto xt:=t^{-1}x$, denoted $(π^{-1},X,T)$. In this paper, we show that $(π,T,X)$ is equicontinuous surjective iff it is uniformly distal iff $(π^{-1},X,T)$ is equicontinuous surjective. As applications of this theorem, we also consider the minimality, distality, and sensitivity of $(π^{-1},X,T)$ if $(π,T,X)$ is invertible with these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of $\mathbb{Z}$-flow with compact zero-dimensional phase space.

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arXiv2018

Uniform positive recursion frequency of any minimal dynamical system on a compact space

Using Gottschalk's notion\,---\,weakly locally almost periodic point, we show in this paper that if $f\colon X\rightarrow X$ is a minimal continuous transformation of a compact Hausdorff space $X$ to itself, then for all entourage $\varepsilon$ of $X$, \begin{equation*} \inf_{x\in X}\left\{\liminf_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}1_{\varepsilon[x]}(f^nx)\right\}>0. \end{equation*} An analogous assertion also holds for each minimal $C^0$-semiflow $π\colon \mathbb{R}_+\times X\rightarrow X$ and for any minimal transformation group with discrete amenable phase group.

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arXiv2018

On transitivity dynamics of topological semiflows

Let $T\times X\rightarrow X, (t,x)\mapsto tx$, be a topological semiflow on a topological space $X$ with phase semigroup $T$. We introduce and discuss in this paper various transitivity dynamics of $(T,X)$.

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arXiv2017

On universal minimal proximal flows of topological groups

In this paper, we show that the action of a characteristically simple, non-extremely amenable (non-strongly amenable, non-amenable) group on its universal minimal (minimal proximal, minimal strongly proximal) flow is effective. We present necessary and sufficient conditions, for the action of a topological group with trivial center on its universal minimal proximal flow, to be effective. A theorem of Furstenberg about the isomorphism of the universal minimal proximal flows of a discrete group and its subgroups of finite index ([Theorem~II.4.4]) is strengthened. Finally, for a pair of groups $H < G$ the same method is applied in order to extend the action of $H$ on its universal minimal proximal flow to an action of its commensurator group $\mathrm{Comm}_G(H)$.

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