arXiv subjects
Publications and source records attributed to Xiongping Dai.
By using ergodic theoretic techniques following Hillel Fürstenberg, we prove that measurable subsets of a locally compact abelian group of positive upper density contain Szemerédi-wise configurations defined by an arbitrary compact subset of the group.
Given any amenable group $G$ (with a left Haar measure $|\cdot|$ or $dg$), we can select out a \textit{Følner subnet} $\{F_θ,θ\inΘ\}$ from any left Følner net in $G$, which is \textit{$L^\infty$-admissible}, namely, for any Borel $G$-space $(X,\mathscr{X})$ and any $φ\in L^\infty(X,\mathscr{X})$, \begin{gather*} \lim_{θ\inΘ}\frac{1}{|F_θ|}\int_{F_θ}φ(gx)dg=φ^*(x)\ \forall x\in X\quad {\textrm{and}}\quad φ^*=(gφ)^*\ \forall g\in G. \end{gather*} Moreover, if $G$ is $σ$-compact such as a locally compact second countable Hausdorff amenable group, then $φ^*\in L^\infty(X,\mathscr{X})$, $φ^*(gx)=φ^*(x)$ \textit{a.e.}, and $φ^*$ is \textit{a.e.} independent of the choice of the admissible Følner net $\{F_θ,θ\inΘ\}$ in $G$. Consequently, we may easily obtain the ergodic disintegration of invariant probability measures for any $σ$-compact amenable group acting Borel on a compact metric space $X$ by continuous transformations of $X$, and the existence of $σ$-finite invariant Radon measures for any Borel action of an amenable group on a locally compact, $σ$-compact, metric space $X$ by continuous maps of $X$, and a $L^\infty$-pointwise multiple ergodic theorem.
Using a recent Furstenberg structure theorem, we obtain a quantitative multiple recurrence theorem relative to any locally compact second countable Noetherian module over a syndetic ring.
Using a recent Furstenberg structure theorem, we obtain Multiple Recurrence Theorems relative to any locally compact second countable Noetherian module $G$ over a syndetic ring $R$, which generalizes Furstenberg's multiple recurrence theorem. In addition we study the multiple Birkhoff center and the pointwise multiple recurrence of a topological $G$-action on a compact metric space.
We study the topological and ergodic dynamics of Bohr almost periodic motions of a topological abelian semigroup acting continuously on a compact metric space.
We extend Furstenberg's structure theorem to any standard Borel $G$-space, where $G$ is any locally compact second countable Noetherian module over a syndetic ring.
In this paper, we study the chaotic dynamics of a minimal center of attraction of a Lagrange stable motion for semi flows.
In this paper, we study the chaotic dynamics of a continuous-time topological semi-flow on a Polish space.
In this paper, we improve the classical multiplicative ergodic theorem.
Quasi-Anosov diffeomorphism is a kind of important dynamical system due to R. Mane 1970s. In this paper, we present a criterion for such dynamics using ergodic theory.
In this paper, we characterize the stability of matrix-valued Markovian chains by periodic data.
In this paper we show that if a linear cocycle is robustly periodical stable then it is uniformly stable.
We study the pointwise stabilizability of a discrete-time, time-homogeneous, and stationary Markovian jump linear system. By using measure theory, ergodic theory and a splitting theorem of state space we show in a relatively simple way that if the system is essentially product-bounded, then it is pointwise convergent if and only if it is pointwise exponentially convergent.
In this paper, we study the fiber-chaos of switched linear dynamical systems.
In this paper, we study the chaotic behavior of a discrete-time linear inclusion.
In this paper, we present some checkable criteria for the spectral finiteness of a finite subset of the real $d\times d$ matrix space $\mathbb{R}^{d\times d}$, where $2\le d<\infty$.
Using a sifting-shadowing combination, we prove in this paper that an arbitrary $\mathrm{C}^1$-class local diffeomorphism $f$ of a closed manifold $M^n$ is uniformly expanding on the closure $\mathrm{Cl}_{M^n}(\mathrm{Per}(f))$ of its periodic point set $\mathrm{Per}(f)$, if it is nonuniformly expanding on $\mathrm{Per}(f)$.
We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant $n$-dimensional subsystems \dot{x}=A_ix+B_i(x)u\quad (x\in\mathbb{R}^n, t\in\mathbb{R}_+ \textrm{and} u\in\mathbb{R}^{m_i}),\qquad \textrm{where} i\in{1,...,N} and a switching signal $σ(\bcdot)\colon\mathbb{R}_+\rightarrow{1,...,N}$ which orchestrates switching between these subsystems above, where $A_i\in\mathbb{R}^{n\times n}, n\ge1, N\ge2, m_i\ge1$, and where $B_i(\bcdot)\colon\mathbb{R}^n\rightarrow\mathbb{R}^{n\times m_i}$ satisfies the condition $\|B_i(x)\|\le\bbbeta\|x\|\;\forall x\in\mathbb{R}^n$. We show that, if ${A_1,...,A_N}$ generates a solvable Lie algebra over the field $\mathbbm{C}$ of complex numbers and there exists an element $\bbA$ in the convex hull $\mathrm{co}{A_1,...,A_N}$ in $\mathbb{R}^{n\times n}$ such that the affine system $\dot{x}=\bbA x$ is exponentially stable, then there is a constant $\bbdelta>0$ for which one can design "sufficiently many" piecewise-constant switching signals $σ(t)$ so that the switching-control systems \dot{x}(t)=A_{σ(t)}x(t)+B_{σ(t)}(x(t))u(t),\quad x(0)\in\mathbb{R}^n\textrm{and} t\in\mathbb{R}_+ are globally exponentially stable, for any measurable external inputs $u(t)\in\mathbb{R}^{m_{σ(t)}}$ with $|u(t)|\le\bbdelta$.