arXiv subjects

Song Shao

Publications and source records attributed to Song Shao.

At least 19 recordsLinked to original sources

arXiv2026

Rigid Functions, IP-Systems, and Topological Mild Mixing

We study uniform rigidity and topological mild mixing through continuous observables. For a fixed sequence of times, the observables rigid along that sequence form a closed unital $T^{\pm1}$-invariant algebra and determine the maximal factor uniformly rigid along the prescribed sequence. We then give functional forms of the classical ${\rm SIP}^{*}$- and ${\rm IP}^{*}$-return-time criteria: a topological dynamical system is mildly mixing exactly when it has no nonconstant locally SIP-rigid observable, and in the minimal category the same property is equivalent to the absence of nonconstant locally IP-rigid observables. Finally, a locally IP-rigid observable yields a canonical orbit-name factor carrying marked local data. For fixed local data, the $T^{\pm 1}$-invariant core of the local rigidity algebra determines a uniformly rigid factor whenever the core is nontrivial.

math.DS↗

arXiv2026

Adherence Semigroups and Density Finite-Sums Configurations

In this paper, we use the adherence semigroup to describe finite-sums configurations in topological dynamical systems. We establish a correspondence between exact finite sumsets and powers of adherence elements. As applications, we give dynamical formulations of density finite-sums results, characterize total minimality through the density of adherence-power orbits, and give a positive answer to the ultrafilter question in [Question~8.9, 6]. We also characterize the sets that belong to the common sum of two commuting nonprincipal ultrafilters.

math.DS↗

arXiv2026

An affirmative answer to Owings's sumset question

We give an affirmative answer to Owings's sumset question: for any $2$-coloring of natural numbers, there is an infinite $B\subseteq\mathbb{N}$ such that $B+B$ is monochromatic. More generally, for every $m,\ell\in\mathbb{N}$ and every $2$-coloring of $\mathbb{N}$, there is an infinite $B\subseteq\mathbb{N}$ such that $$ (m+\ell)B\cup\{mx+\ell y:x,y\in B,\ x<y\} $$ is monochromatic.

math.CO↗

arXiv2025

Context-measure: Contextualizing Metric for Camouflage

Camouflage relies heavily on context, but current metrics used in camouflaged object segmentation ignore contextual cues. We identify two major drawbacks of these metrics: first, the Dimension Flaw - a predicted foreground map usually contains both pixel labels and probability scores, whereas ground truth provides only one-dimensional binary labels; second, the Range Flaw - these metrics struggle to capture full-range pixel dependencies. Thus, we propose Context-measure, a novel context-aware evaluation paradigm built on a probabilistic pixel correlation framework. It augments the ground truth with pixel-level contextual affinity and builds a perception cycle, achieving greater consistency with human perception. Extensive experiments using four meta-measures show that our Context-measure comprehensively outperforms all widely adopted metrics for camouflaged object segmentation. To our knowledge, this is the first metric designed for camouflaged scenarios. Code is available at https://github.com/pursuitxi/Context-measure.

cs.CV↗

arXiv2025

Structure theorems of commuting transformations and minimal $\mathbb{R}$-flows

In this paper, we develop several structure theorems concerning commuting transformations and minimal $\mathbb{R}$-flows. Specifically, we show that if $(X,S)$, $(X,T)$ are minimal systems with $S$ and $T$ being commutative, then they possess an identical higher-order regionally proximal relation. Consequently, both $(X, S)$ and $(X, T)$ share the same increasing sequence of pro-nilfactors. For minimal $\mathbb{R}$-flows, we introduce the concept of higher-order regionally proximal relations and nilfactors, and establish that nilfactors are characteristic factors for minimal $\mathbb{R}$-flows, up to almost one to one extensions.

math.DS↗

arXiv2025

On systems disjoint from all minimal systems

Recently, G\'{o}rska, Lema\'{n}czyk, and de la Rue characterized the class of automorphisms disjoint from all ergodic automorphisms. Inspired by their work, we provide several characterizations of systems that are disjoint from all minimal systems. For a topological dynamical system $(X,T)$, it is disjoint from all minimal systems if and only if there exist minimal subsets $(M_i)_{i\in\mathbb{N}}$ of $X$ whose union is dense in $X$ and each of them is disjoint from $X$ (we also provide a measure-theoretical analogy of the result). For a semi-simple system $(X,T)$, it is disjoint from all minimal systems if and only if there exists a dense $G_{\delta}$ set $\Omega$ in $X \times X$ such that for every pair $(x_1,x_2) \in \Omega$, the subsystems $\overline{\mathcal{O}}(x_1,T)$ and $\overline{\mathcal{O}}(x_2,T)$ are disjoint. Furthermore, for a general system a characterization similar to the ergodic case is obtained.

math.DS↗

arXiv2024

Multiple recurrence without commutativity

We study multiple recurrence without commutativity in this paper. We show that for any two homeomorphisms $T,S: X\rightarrow X$ with $(X,T)$ and $(X,S)$ being minimal, there is a residual subset $X_0$ of $X$ such that for any $x\in X_0$ and any nonlinear integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, there is some subsequence $\{n_i\}$ of $\mathbb Z$ with $n_i\to \infty$ satisfying $$ S^{n_i}x\to x,\ T^{p_1(n_i)}x\to x, \ldots,\ T^{p_d(n_i)}x\to x,\ i\to\infty.$$

math.DS↗

arXiv2024

Monochromatic Sums and Products with Additive or Multiplicative Shifts in Natural Numbers

In this paper we prove that for any finite coloring of N there are lambda,rho in N such that infinitely many pairs (x,y),(u,v) in N^2 satisfy the sets {lambda x, lambda y, x y, lambda(x+y)} and {u+rho, v+rho, u v+rho, u+v} being monochromatic. Using related arguments we also give two different proofs of a special case of the Milliken--Taylor theorem.

math.CO↗

arXiv2024

A counterexample on multiple convergence without commutativity

It is shown that there exist a probability space $(X,{\mathcal X},\mu)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\mathcal X},\mu)$ with $h_\mu(X,T)=h_\mu(X,S)=0$, and $f, g \in L^\infty(X,\mu)$ such that the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x) \end{equation*} does not exist in $L^2(X,\mu)$.

math.DS↗

arXiv2023

Pointwise convergence of some continuous-time polynomial ergodic averages

In this paper, we study the pointwise convergence of centain continuous-time polynomial ergodic averages. Our approach is based on the topological models of measurable flows. One of the main results of this paper is as follows: Let $a\in \mathbb{R}$, $Q\in \mathbb{R}[t]$ with $\text{deg}\ Q\ge 2$. Let $(X,\mathcal{X},\mu, (T^{t})_{t\in \mathbb{R}})$ and $(X,\mathcal{X},\mu, (S^{t})_{t\in \mathbb{R}})$ be two measurable flows. Then for any $f_1, f_2, g\in L^{\infty}(\mu)$, the limit \begin{equation*} \lim\limits_{M\to\infty}\frac{1}{M}\int_{0}^{M}f_1(T^{t}x)f_2(T^{at}x)g(S^{Q(t)}x)dt \end{equation*} exists for $\mu$-a.e. $x\in X$. In particular, we are able to build a pointwise ergodic theorem involving geodesic flow and horocycle flow.

math.DS↗

arXiv2023

A counterexample on polynomial multiple convergence without commutativity

It is shown that for polynomials $p_1, p_2 \in {\mathbb Z}[t]$ with ${\rm deg}\ p_1, {\rm deg}\ p_2\ge 5$ there exist a probability space $(X,{\mathcal X},\mu)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\mathcal X},\mu)$ with $h_\mu(X,T)=h_\mu(X,S)=0$, and $f, g \in L^\infty(X,\mu)$ such that the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{p_1(n)}x)g(S^{p_2(n)}x) \end{equation*} does not exist in $L^2(X,\mu)$, which in some sense answers a question by Frantzikinakis and Host.

math.DS↗

arXiv2023

Topological dynamical systems induced by polynomials and combinatorial consequences

Let $d\in {\mathbb N}$ and $p_i$ be an integral polynomial with $p_i(0)=0$, $1\le i\le d$. It is shown that if $S$ is piecewise syndetic in $\mathbb Z$, then $$\{(m,n)\in{\mathbb Z}^2: m+p_1(n),\ldots,m+p_d(n)\in S\}$$ is piecewise syndetic in ${\mathbb Z}^2$, which extends the result by Glasner and Furstenberg for linear polynomials. Our result is obtained by showing the density of minimal points of a dynamical system of ${\mathbb Z}^2$ action associated with the piecewise syndetic set $S$ and the polynomials $\{p_1,\ldots,p_d\}$. Moreover, it is proved that if $(X,T)$ is minimal, then for each non-empty open subset $U$ of $X$, there is $x\in U$ with $\{n\in {\mathbb Z}: T^{p_1(n)}x\in U, \ldots, T^{p_d(n)}x\in U\}$ piecewise syndetic.

math.DS↗

arXiv2023

Polynomial Furstenberg joinings and its applications

In this paper, a polynomial version of Furstenberg joining is introduced and its structure is investigated. Particularly, it is shown that if all polynomials are non-linear, then almost every ergodic component of the joining is a direct product of an infinity-step pro-nilsystem and a Bernoulli system. As applications, some new convergence theorems are obtained. Particularly, it is proved that if $T$ and $S$ are ergodic measure preserving transformations on a probability space $(X,{\mathcal X},\mu)$ and $T$ has zero entropy, then for all $c_i\in {\mathbb Z}\setminus \{0\}$, all integral polynomials $p_j$ with $\deg {p_j}\ge 2$, and for all $f_i, g_j\in L^\infty(X,\mu)$, $1\le i\le m$ and $1\le j\le d$, $$\lim_{N\to\infty} \frac{1}{N}\sum_{n=0}^{N-1}f_1(T^{c_1n}x)\cdots f_m(T^{c_mn}x)\cdot g_1(S^{p_1(n)}x)\cdots g_d(S^{p_d(n)}x),$$ exists in $L^2(X,\mu)$, which extends the recent result by Host and Frantzikinakis. Moreover, it is shown that for an ergodic measure-preserving system $(X,{\mathcal X},\mu,T)$, a non-linear integral polynomial $p$ and $f\in L^\infty(X,\mu)$, the Furstenberg systems of $\big(f(T^{p(n)})x\big)_{n\in {\mathbb Z}}$ are ergodic and isomorphic to direct products of infinite-step pro-nilsystems and Bernoulli systems for almost every $x\in X$, which answers a problem by Frantzikinakis.

math.DS↗

arXiv2021

Almost proximal extensions of minimal flows

In this paper we study almost proximal extensions of minimal flows. Let $\pi: (X,T)\rightarrow (Y,T)$ be an extension of minimal flows. $\pi$ is called an almost proximal extension if there is some $N \in \mathbb{N}$ such that the cardinality of any almost periodic subset in each fiber is not greater than $N$. When $N=1$, $\pi$ is proximal. We will give the structure of $\pi$ and give a dichotomy theorem: any almost proximal extension of minimal flows is either almost finite to one, or almost all fibers contain an uncountable strongly scrambled subset. Using category method Glasner and Weiss showed the existence of proximal but not almost one to one extensions [18]. In this paper, we will give explicit such examples, and also examples of almost proximal but not almost finite to one extensions.

math.DS↗

arXiv2021

Multiply minimal points for the product of iterates

The multiple Birkhoff recurrence theorem states that for any $d\in\mathbb N$, every system $(X,T)$ has a multiply recurrent point $x$, i.e. $(x,x,\ldots, x)$ is recurrent under $\tau_d=:T\times T^2\times \ldots \times T^d$. It is natural to ask if there always is a multiply minimal point, i.e. a point $x$ such that $(x,x,\ldots,x)$ is $\tau_d$-minimal. A negative answer is presented in this paper via studying the horocycle flows. However, it is shown that for any minimal system $(X,T)$ and any non-empty open set $U$, there is $x\in U$ such that $\{n\in{\mathbb Z}: T^nx\in U, \ldots, T^{dn}x\in U\}$ is piecewise syndetic; and that for a PI minimal system, any $M$-subsystem of $(X^d, \tau_d)$ is minimal.

math.DS↗

arXiv2021

Topological mild mixing of all orders along polynomials

A minimal system $(X,T)$ is topologically mildly mixing if all non-empty open subsets $U,V$, $\{n\in \Z: U\cap T^{-n}V\neq \emptyset\}$ is an IP$^*$-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that $(X,T)$ is a topologically mildly mixing minimal system, $d\in \N$, $p_1(n),\ldots, p_d(n)$ are integral polynomials with no $p_i$ and no $p_i-p_j$ constant, $1\le i\neq j\le d$, then for all non-empty open subsets $U , V_1, \ldots, V_d $, $\{n\in \Z: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}V_2\cap \ldots \cap T^{-p_d(n) }V_d \neq \emptyset \}$ is an IP$^*$-set. We also give the theorem for systems under abelian group actions.

math.DS↗

arXiv2020

Topological characteristic factors and nilsystems

We prove that the maximal infinite step pro-nilfactor $X_\infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $π:X \rightarrow X_\infty$, the induced open extension $π^*:X^* \rightarrow X^*_\infty$ has the following property: for $x$ in a dense $G_δ$ set of $X^*$, the orbit closure $L_x=\overline{\mathcal{O}}((x,x,\ldots,x), T\times T^2\times \ldots \times T^d)$ is $(π^*)^{(d)}$-saturated, i.e. $L_x=((π^*)^{(d)})^{-1}(π^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $k\ge 2$, then for any $d\in {\mathbb N}$ and any $0\le j<k$ there is a sequence $\{n_i\}$ of $\mathbb Z$ with $n_i\equiv j\ (\text{mod}\ k)$ such that $T^{n_i}x\rightarrow x, T^{2n_i}x\rightarrow x, \ldots, T^{dn_i}x\rightarrow x$ for $x$ in a dense $G_δ$ subset of $X$; (2) if $(X,T)$ is totally minimal, then $\{T^{n^2}x:n\in {\mathbb Z}\}$ is dense in $X$ for $x$ in a dense $G_δ$ subset of $X$; (3) for any $d\in\mathbb N$ and any minimal system, which is an open extension of its maximal distal factor, ${\bf RP}^{[d]}={\bf AP}^{[d]}$, where the latter is the regionally proximal relation of order $d$ along arithmetic progressions.

math.DS↗

arXiv2020

Topological characteristic factors and independence along arithmetic progressions

Let $π: (X,T)\rightarrow (Y,T)$ be a factor map of topological dynamics and $d\in {\mathbb {N}}$. $(Y,T)$ is said to be a $d$-step topological characteristic factor if there exists a dense $G_δ$ set $X_0$ of $X$ such that for each $x\in X_0$ the orbit closure $\overline{\mathcal O}((x, \ldots,x), T\times T^2\times \ldots \times T^d)$ is $π\times \ldots \times π$ ($d$ times) saturated. In 1994 Eli Glasner studied the topological characteristic factor for minimal systems. For example, it is shown that for a distal minimal system, its largest distal factor of order $d-1$ is its $d$-step topological characteristic factor. In this paper, we generalize Glasner's work to the product system of finitely many minimal systems and give its relative version. To prove these results, we need to deal with $(X,T^m)$ for $m\in {\mathbb {N}}$. We will study the structure theorem of $(X,T^m)$. We show that though for a minimal system $(X,T)$ and $m\in {\mathbb {N}}$, $(X,T^m)$ may not be minimal, but we still can have PI-tower for $(X,T^m)$ and in fact it looks the same as the PI tower of $(X,T)$. We give some applications of the results developed. For example, we show that if a minimal system has no nontrivial independent pair along arithmetic progressions of order $d$, then up to a canonically defined proximal extension, it is PI of order $d$; if a minimal system $(X,T)$ has a nontrivial $d$-step topological characteristic factor, then there exist ``many'' $Δ$-transitive sets of order $d$.

math.DS↗