arXiv subjects

Jiahao Qiu

Publications and source records attributed to Jiahao Qiu.

39 records · Page 3Linked to original sources

arXiv2021

Independence and almost automorphy of high order

In this paper, it is shown that for a minimal system $(X,T)$ and $d,k\in \mathbb{N}$, if $(x,x_i)$ is regionally proximal of order $d$ for $1\leq i\leq k$, then $(x,x_1,\ldots,x_k)$ is $(k+1)$-regionally proximal of order $d$. Meanwhile, we introduce the notion of $\mathrm{IN}^{[d]}$-pair: for a dynamical system $(X,T)$ and $d\in \mathbb{N}$, a pair $(x_0,x_1)\in X\times X$ is called an $\mathrm{IN}^{[d]}$-pair if for any $k\in \mathbb{N}$ and any neighborhoods $U_0 ,U_1 $ of $x_0$ and $x_1$ respectively, there exist integers $p_j^{(i)},1\leq i\leq k,$ $1\leq j\leq d$ such that $$ \bigcup_{i=1}^k\{ p_1^{(i)}ε(1)+\ldots+p_d^{(i)} ε(d):ε(j)\in \{0,1\},1\leq j\leq d\}\backslash \{0\}\subset \mathrm{Ind}(U_0,U_1), $$ where $\mathrm{Ind}(U_0,U_1)$ denotes the collection of all independence sets for $(U_0,U_1)$. It turns out that for a minimal system, if it dose not contain any nontrivial $\mathrm{IN}^{[d]}$-pair, then it is an almost one-to-one extension of its maximal factor of order $d$.

math.DS↗

arXiv2020

Pro-nilfactors of the space of arithmetic progressions in topological dynamical systems

For a topological dynamical system $(X, T)$, $l\in\mathbb{N}$ and $x\in X$, let $N_l(X)$ and $L_x^l(X)$ be the orbit closures of the diagonal point $(x,x,\ldots,x)$ ($l $ times) under the actions $\mathcal{G}_{l}$ and $τ_l $ respectively, where $\mathcal{G}_{l}$ is generated by $T\times T\times \ldots \times T$ ($l $ times) and $τ_l=T\times T^2\times \ldots \times T^l$. In this paper, we show that for a minimal system $(X,T)$ and $l\in \mathbb{N}$, the maximal $d$-step pro-nilfactor of $(N_l(X),\mathcal{G}_{l})$ is $(N_l(X_d),\mathcal{G}_{l})$, where $π_d:X\to X/\mathbf{RP}^{[d]}=X_d,d\in \mathbb{N}$ is the factor map and $\mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$. Meanwhile, when $(X,T)$ is a minimal nilsystem, we also calculate the pro-nilfactors of $(L_x^l(X),τ_l)$ for almost every $x$ w.r.t. the Haar measure. In particular, there exists a minimal $2$-step nilsystem $(Y,T)$ and a countable set $Ω\subset Y$ such that for $y\in Y\backslash Ω$ the maximal equicontinuous factor of $(L_y^2(Y),τ_2)$ is not $(L_{π_1(y)}^2(Y_{1}),τ_2)$.

math.DS↗

arXiv2018

A note on mean equicontinuity

In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system.

math.DS↗