arXiv subjects
Publications and source records attributed to Enhui Shi.
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.
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 \[\liminf_{n\rightarrow\infty} r\left((2n+1)^2, q(n)\right)<\infty,\] where $r(\ ,\ )$ is the opposite-Ramsey number function.
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 rational numbers to obtain a measure rigidity result for endomorphisms on $n$-torus.
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 sensitive action of a group $G$, then $G$ contains a free subsemigroup and the action has positive geometric entropy. As a corollary, $X$ admits no sensitive nilpotent group action.
We show that if $G$ is a countable amenable group acting on a uniquely arcwise connected continuum $X$, then $G$ has either a fixed point or a 2-periodic point in $X$.
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.
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 subgroup $H$ of $G$ with $[G: H]=\infty$ has this property; is almost minimal if it has at most countably many nontransitive points. In the paper, we determine all the topological conjugation classes of tightly transitive and almost minimal subgroups of $\text{Homeo}_{+}(\mathbb{S}^1)$ which are isomorphic to $\mathbb{Z}^n$ for any integer $n\geq 2$.
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 $\mathbb R$; (3) nonexistence of infinite towers covering the whole line. For a nilpotent subgroup $G$ of $\text{Homeo}_+(\mathbb{R})$, we show that $G$ always has an invariant Radon measure and a minimal closed set if every element of $G$ is $C^{1+α} (α>0$); a counterexample of $C^1$ commutative subgroup of $\text{Homeo}_+(\mathbb{R})$ is constructed.
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 particular, if $X$ is homotopically equivalent to a CW complex, then $X$ cannot be simply connected.
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 $\mathcal S$.
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 we introduce a classification theorem for a specified class of topologically transitive orientation-preserving group actions on the circle by $\mathbb Z^d$. Also, some groups that admit/admit no topologically transitive actions on the line are determined.
Let $d\geq 2$ be an integer and let $ω_1,\cdots ,ω_d$ be moduli of continuity in a specified class which contains the moduli of Hölder continuity. Let $f_k$, $k\in\{1,\cdots,d\}$, be $C^{1+ω_k}$ orientation preserving diffeomorphisms of the circle and $f_1,\cdots, f_d$ commute with each other. We prove that if the rotation numbers of $f_k$'s are independent over the rationals and $ω_1(t)\cdotsω_d(t)=tω(t)$ with $\lim_{t\rightarrow 0^+}ω(t)=0$, then $f_1,\cdots,f_d$ are simultaneously (topologically) conjugate to rigid rotations.
We show that for any finite coloring of the group $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ and for any positive integer $k$, there always exists a monochromatic regular $k$-gon in $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ with respect to the word length metric induced by the standard generating set; the edge length of which is estimated.
We introduce the notion of strongly independent matrices and show the existence of strongly independent matrices in $GL(n,\mathbb{Z})$ over $\mathbb{Z}^n\setminus\{0\}$ when $2n+1$ is a prime number. As an application of strong independence, we give a measure rigidity result for endomorphisms on $n$-torus $\mathbb{T}^n$.
For every group $G$, we show that either $G$ has a topologically transitive action on the line $\mathbb R$ by orientation-preserving homeomorphisms, or every orientation-preserving action of $G$ on $\mathbb R$ has a wandering interval. According to this result, all groups are divided into two types: transitive type and wandering type, and the types of several groups are determined. We also show that every finitely generated orderable group of wandering type is indicable. As a corollary, we show that if a higher rank lattice $Γ$ is orderable, then $Γ$ is of transitive type.
Let $X$ be a Peano continuum having a free arc and let $C^0(X)$ be the semigroup of continuous self-maps of $X$. A subsemigroup $F\subset C^0(X)$ is said to be sensitive, if there is some constant $c>0$ such that for any nonempty open set $U\subset X$, there is some $f\in F$ such that the diameter ${\rm diam}(f(U))>c$. We show that if $X$ admits a sensitive commutative subsemigroup $F$ of $C^0(X)$ consisting of continuous open maps, then either $X$ is an arc, or $X$ is a circle.
The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group $\H$. We establish the decomposition of the tangent space of any $C^\infty$ compact Riemannian manifold $M$ for Lyapunov exponents, and show that all Lyapunov exponents for the center elements are zero. We obtain that if an $\H$ group action contains an Anosov element, then under certain conditions on the element, the center elements are of finite order. In particular there is no faithful codimensional one Anosov Heisenberg group action on any manifolds, and no faithful codimensional two Anosov Heisenberg group action on tori. In addition, we show smooth local rigidity for higher rank ergodic $\H$ actions by toral automorphisms, using a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme.