唯一弧连通局部弧连通空间上的点态递归映射

Periodic points for amenable group actions on uniquely arcwise connected continua

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 $X$.

Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point

We are dealing with the question whether every group or semigroup action (with some additional property) on a continuum (with some additional property) has a fixed point. One of such results was given in 2009 by Shi and Sun. They proved that every nilpotent group action on a uniquely arcwise connected continuum has a fixed point. We are seeking for this type of results with e.g. commutative, compact or torsion groups and semigroups acting on dendrites, dendroids, $λ$-dendroids and uniquely arcwise connected continua. We prove that every continuous action of a compact or torsion group on a uniq…

Rigidity of matrix group actions on CAT(0) spaces with possible parabolic isometries and uniquely arcwise connected spaces

It is well-known that $\mathrm{SL}_{n}(\mathbf{Q}_{p})$ acts without fixed points on an $(n-1)$-dimensional $\mathrm{CAT}(0)$ space (the affine building). We prove that $n-1$ is the smallest dimension of $\mathrm{CAT}(0)$ spaces on which matrix groups act without fixed points. Explicitly, let $R$ be an associative ring with identity and $E_{n}^{\prime }(R)$ the extended elementary subgroup. Any isometric action of $E_{n}^{\prime }(R)$ on a complete $\mathrm{CAT(0)}$ space $X^{d}$ of dimension $d<n-1$ has a fixed point. Similar results are discussed for automorphism groups of free groups. Furth…

局部紧空间上半群作用的逐点连续遍历性

Continuous pointwise ergodicity for semigroup actions on locally compact spaces

该研究针对局部紧空间上的半群作用，建立了逐点连续遍历性的算子理论刻画，推广了紧空间群作用的经典结果，且适用于无Følner序列的半群，还给出了相关动力学实例。