Related papers: Minimal sets and orbit space for group actions on …

Consider a smooth action of $\mathbb R^n$ on a connected manifold $M$, not necessarily compact, of dimension $m$ and rank $k$. Assume that $M$ is not a cylinder. Then there exists an orbit of the action of dimension $<(m+k)/2$. As a…

Given a finite group $G$, the solubilizer of an element $x$, denoted by $\Sol_G(x)$, is the set of all elements $y$ such that $\langle x, y\rangle$ is a soluble subgroup of $G$. In this paper, we provide a classification for all…

Let G be a semi-simple Lie group and Q a parabilic subgroup of its complexification G^\mathbb C, then Z:=G^\mathbb C/Q is a compact complex homogeneous manifold. Moreover, G as well as K^\mathbb C, the complexification of the maximal…

Let $G$ be second countable locally compact Hausdorff groupoid with a continuous Haar system. We remove the assumption of amenability in a theorem by Clark about GCR groupoid $C^*$-algebras. We show that if the groupoid $C^*$-algebra of $G$…

We demonstrate that And\'ujar Guerrero, Thomas and Walsberg's results on definable compactness in o-minimal structures still hold true in definably complete locally o-minimal structures. As an application, we show that a definably simple…

Let $k$ be a finitely generated field, let $X$ be an algebraic variety and $G$ a linear algebraic group, both defined over $k$. Suppose $G$ acts on $X$ and every element of a Zariski-dense semigroup $\Gamma \subset G(k)$ has a rational…

Fix a module M over a local ring R and a group action G on M, not necessarily R-linear. To understand how large is the G-orbit of an element z\in M one looks for the large submodules of M lying in Gz. We provide the corresponding…

In this paper, the study of the global orbit pattern (gop) formed by all the periodic orbits of discrete dynamical systems on a finite set $X$ allows us to describe precisely the behaviour of such systems. We can predict by means of closed…

Let $G$ be a totally disconnected, locally compact (t.d.l.c.) group. The scale $s_G(g)$ of $g \in G$ in the sense of Willis is given by the minimum value of the index $|gUg^{-1}:U \cap gUg^{-1}|$ as $U$ ranges over the compact open…

Let $\beta: S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that the crossed product $C(S^{2n+1})\rtimes_{\beta} \Z$ has rational tracial rank at most one. More generally, let $\Omega$ be a connected compact metric…

Let $\Gamma\curvearrowright (X,\mu)$ be a measure preserving action of a countable group $\Gamma$ on a standard probability space $(X,\mu)$. We prove that if the action $\Gamma\curvearrowright X$ is not profinite and satisfies a certain…

We say that an element $g$ of a group $G$ is almost right Engel if there is a finite set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$, that is, for…

Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. It is shown that the universal minimal space M(G) of the topological group G=Homeo(X), is the space of maximal…

We prove universal lower bounds for discrepancies (i.e. sizes of spectral gaps of averaging operators) of measure-preserving actions of a locally compact group on probability spaces. For example, a locally compact Hausdorff unimodular group…

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to…

A graph $G$ is a DTDP-graph if it has a pair $(D,T)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $T$ is a total dominating set of $G$. Such graphs were studied in a number of research papers. In this paper we…

Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F of X such that FG=X and…

Let $\mathbb{G}$ be a compact Hausdorff group acting on a compact Hausdorff space $X$, $\alpha$ an irreducible $\mathbb{G}$-representation, and $C(X)$ the $C^*$-algebra of complex-valued continuous functions on $X$. We prove that the…

It is known that if each point $x$ of a dynamical system is generic for some invariant measure $\mu_x$, then there is a strong connection between certain ergodic and topological properties of that system. In particular, if the acting group…

Let $G$ be a noncompact real algebraic group and $\G<G$ a lattice. One purpose of this paper is to show that there is an smooth, volume preserving, mixing action of $G$ or $\G$ on a compact manifold which admits a smooth deformation. We…