Enhui Shi
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View article: A note on the bounded orbit conjecture
A note on the bounded orbit conjecture Open
If $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is an orientation reversing fixed point free homeomorphism on the plane $\mathbb{R}^2$ with no unbounded orbit, then $f$ has infinitely many periodic orbits.
View article: Linearizations of periodic point free distal homeomorphisms on the annulus
Linearizations of periodic point free distal homeomorphisms on the annulus Open
Let $\mathbb{A}$ be an annulus in the plane $\mathbb R^2$ and $g:\mathbb{A}\rightarrow \mathbb{A}$ be a boundary components preserving homeomorphism which is distal and has no periodic points. In \cite{SXY}, the authors show that there is …
View article: $(ω, α, n)$-sensitivity and limit sets of zero entropy homeomorphisms on the square
$(ω, α, n)$-sensitivity and limit sets of zero entropy homeomorphisms on the square Open
For a homeomorphism $f$ of a compact metric space $X$ and a positive integer $n\geq 2$, we introduce the notion of $(ω, α, n)$-sensitivity of $f$, which describes such a kind of chaos: there is some $c>0$ such that for any $x\in X$ and any…
View article: A new construction of counterexamples to the bounded orbit conjecture
A new construction of counterexamples to the bounded orbit conjecture Open
The bounded orbit conjecture says that every homeomorphism on the plane with each of its orbits being bounded must have a fixed point. Brouwer's translation theorem asserts that the conjecture is true for orientation preserving homeomorphi…
View article: The structure of periodic point free distal homeomorphisms on the annulus
The structure of periodic point free distal homeomorphisms on the annulus Open
Let $A$ be an annulus in the plane $\mathbb R^2$ and $g:A\rightarrow A$ be a boundary components preserving homeomorphism which is distal and has no periodic points. Then there is a continuous decomposition of $A$ into $g$-invariant circle…
View article: Can points of bounded orbits surround points of unbounded orbits ?
Can points of bounded orbits surround points of unbounded orbits ? Open
We show a somewhat surprising result: if $E$ is a disk in the plane $\mathbb R^2$, then there is a homeomorphism $h:\mathbb R^2\rightarrow\mathbb R^2$ such that, for every $x\in\partial E$, the orbit $O(x, h)$ is bounded, but for every $y\…
View article: Some extensions of the Brouwer fixed point theorem
Some extensions of the Brouwer fixed point theorem Open
We study the existence of fixed points for continuous maps $f$ from an $n$-ball $X$ in $\mathbb R^n$ to $\mathbb R^n$ with $n\geq 1$. We show that $f$ has a fixed point if, for some absolute retract $Y\subset\partial X$, $f(Y)\subset X$ an…
View article: Structures of $R(f)-\overline{P(f)}$ for graph maps $f$
Structures of $R(f)-\overline{P(f)}$ for graph maps $f$ Open
Let $G$ be a graph and $f: G\rightarrow G$ be a continuous map. We establish a structure theorem which describes the structures of the set $R(f)-\overline{P(f)}$, where $R(f)$ and $P(f)$ are the recurrent point set and the periodic point s…
View article: Groups acting distally and minimally on $\mathbb S^2$ and $\mathbb {RP}^2$
Groups acting distally and minimally on $\mathbb S^2$ and $\mathbb {RP}^2$ Open
Let $X$ be the $2$-sphere $\mathbb S^2$ or the real projective plane $\mathbb {RP}^2$. We show that if $Γ$ is a finitely generated group acting minimally and distally on $X$, then $Γ$ contains a nonabelian free subgroup.
View article: The structures of pointwise recurrent quasi-graph maps
The structures of pointwise recurrent quasi-graph maps Open
We show that a continuous map $f$ from a quasi-graph $G$ to itself is pointwise recurrent if and only if one of the following two statements holds: (1) $X$ is a simple closed curve and $f$ is topologically conjugate to an irrational rotati…
View article: The structures of higher rank lattice actions on dendrites
The structures of higher rank lattice actions on dendrites Open
Let $Γ$ be a higher rank lattice acting on a nondegenerate dendrite $X$ with no infinite order points. We show that there exists a nondegenerate subdendrite $Y$ which is $Γ$-invariant and satisfies the following items: (1) There is an inve…
View article: Rigidity for higher rank lattice actions on dendrites
Rigidity for higher rank lattice actions on dendrites Open
We study the rigidity in the sense of Zimmer for higher rank lattice actions on dendrites and show that: (1) if $Γ$ is a higher rank lattice and $X$ is a nondegenerate dendrite with no infinite order points, then any action of $Γ$ on $X$ c…
View article: The nonexistence of expansive polycyclic group actions on the circle $\mathbb S^1$
The nonexistence of expansive polycyclic group actions on the circle $\mathbb S^1$ Open
We show that the circle $\mathbb S^1$ admits no expansive polycyclic group actions.
View article: The structure of pointwise recurrent expansive homeomorphisms
The structure of pointwise recurrent expansive homeomorphisms Open
Let $X$ be a compact metric space and let $f:X\rightarrow X$ be a homeomorphism on $X$. We show that if $f$ is both pointwise recurrent and expansive, then the dynamical system $(X, f)$ is topologically conjugate to a subshift of some symb…
View article: The non-coexistence of distality and expansivity for group actions on infinite compacta
The non-coexistence of distality and expansivity for group actions on infinite compacta Open
Let $X$ be a compact metric space and $G$ a finitely generated group. Suppose $ϕ:G\rightarrow {\rm Homeo}(X)$ is a continuous action. We show that if $ϕ$ is both distal and expansive, then $X$ must be finite. A counterexample is constructe…
View article: An alternative for minimal group actions on totally regular curves
An alternative for minimal group actions on totally regular curves Open
Let $G$ be a countable group and $X$ be a totally regular curve. Suppose that $ϕ:G\rightarrow {\rm Homeo}(X)$ is a minimal action. Then we show an alternative: either the action is topologically conjugate to isometries on the circle $\math…
View article: The nonexistence of expansive actions on Suslinian continua by groups of subexponential growth
The nonexistence of expansive actions on Suslinian continua by groups of subexponential growth Open
We show that if $G$ is a finitely generated group of subexponential growth and $X$ is a Suslinian continuum, then any action of $G$ on $X$ cannot be expansive.
View article: A dynamical argument for a Ramsey property
A dynamical argument for a Ramsey property Open
We show by a dynamical argument that there is a positive integer valued function $q$ defined on positive integer set $\mathbb N$ such that $q([\log n]+1)$ is a super-polynomial with respect to positive $n$ and \[\liminf_{n\rightarrow\infty…
View article: Sensitive group actions on regular curves of almost $\leq n$ order
Sensitive group actions on regular curves of almost $\leq n$ order Open
Let $X$ be a regular curve and $n$ be a positive integer such that for every nonempty open set $U\subset X$, there is a nonempty connected open set $V\subset U$ with the cardinality $|\partial_X(V)|\leq n$. We show that if $X$ admits a sen…
View article: Periodic points for amenable group actions on uniquely arcwise connected continua
Periodic points for amenable group actions on uniquely arcwise connected continua Open
We show that any action of a countable amenable group on a uniquely arcwise connected continuum has a periodic point of order $\leq 2$ .
View article: Continua having distal minimal actions by amenable groups
Continua having distal minimal actions by amenable groups Open
Let $X$ be a non-degenerate connected compact metric space. If $X$ admits a distal minimal action by a finitely generated amenable group, then the first \vCech cohomology group $ {\check H}^1(X)$ with integer coefficients is nontrivial. In…
View article: Distal higher rank lattice actions on surfaces
Distal higher rank lattice actions on surfaces Open
Let $Γ$ be a lattice in ${\rm SL}(n, \mathbb R)$ with $n\geq 3$ and $\mathcal S$ be a closed surface. Then $Γ$ has no distal minimal action on $\mathcal S$.
View article: Multilabel Feature Selection Using Mutual Information and ML-ReliefF for Multilabel Classification
Multilabel Feature Selection Using Mutual Information and ML-ReliefF for Multilabel Classification Open
Recently, multilabel classification algorithms play an increasingly significant role in data mining and machine learning. However, some existing mutual information-based algorithms ignore the influence of the proportions of labels on the c…
View article: Equicontinuity of minimal sets for amenable group actions on dendrites
Equicontinuity of minimal sets for amenable group actions on dendrites Open
In this note, we show that if $G$ is an amenable group acting on a dendrite $X$, then the restriction of $G$ to any minimal set $K$ is equicontinuous, and $K$ is either finite or homeomorphic to the Cantor set.
View article: Invariant Radon measures and minimal sets for subgroups of $\text{Homeo}_+(\mathbb{R})$
Invariant Radon measures and minimal sets for subgroups of $\text{Homeo}_+(\mathbb{R})$ Open
Let $G$ be a subgroup of $\text{Homeo}_+(\mathbb{R})$ without crossed elements. We show the equivalence among three items: (1) existence of $G$-invariant Radon measures on $\mathbb R$; (2) existence of minimal closed subsets of $\mathbb R$…
View article: The realization and classification of topologically transitive group actions on $1$-manifolds
The realization and classification of topologically transitive group actions on $1$-manifolds Open
In this report, we first recall the Poincaré's classification theorem for minimal orientation-preserving homeomorphisms on the circle and the Ghys' classification theorem for minimal orientation-preserving group actions on the circle. Then…
View article: Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus
Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus Open
We introduce the concept of strongly independent matrices over any field, and prove the existence of such matrices for certain fields and the non-existence for algebraically closed fields. Then we apply strongly independent matrices over r…
View article: Topological conjugation classes of tightly transitive subgroups of $\text{Homeo}_{+}(\mathbb{S}^1)$
Topological conjugation classes of tightly transitive subgroups of $\text{Homeo}_{+}(\mathbb{S}^1)$ Open
Let $\text{Homeo}_{+}(\mathbb{S}^1)$ denote the group of orientation preserving homeomorphisms of the circle $\mathbb{S}^1$. A subgroup $G$ of $\text{Homeo}_{+}(\mathbb{S}^1)$ is tightly transitive if it is topologically transitive and no …