Related papers: A note on the surjunctivity of algebraic dynamical…

Let $\Gamma$ be a simple connect graph on a finite vertex set $V$ and let $A$ be its adjacency matrix. Then $\Gamma$ is said to be \textit{singular} if and only if $0$ is an eigenvalue of $A.$ The \textit{nullity (singularity)} of $\Gamma,$…

Consider a free ergodic measure preserving profinite action $\Gamma\curvearrowright X$ (i.e. an inverse limit of actions $\Gamma\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally of a group…

In this paper we examine the interplay between recurrence properties and the shadowing property in dynamical systems on compact metric spaces. In particular, we demonstrate that if the dynamical system $(X,f)$ has shadowing, then it is…

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application…

For every finite-to-one map $\lambda:\Gamma\to\Gamma$ and for every abelian group $K$, the generalized shift $\sigma_\lambda$ of the direct sum $\bigoplus_\Gamma K$ is the endomorphism defined by…

The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics. We study dynamical properties of a larger class of maps. Our main result is that, under some natural conditions, a map of this class has a completely…

A group $G$ of permutations of a set $\Omega$ is {\em primitive} if it acts transitively on $\Omega$, and the only $G$-invariant equivalence relations on $\Omega$ are the trivial and universal relations. A graph $\Gamma$ is {\em primitive}…

Let $M$ be a closed, oriented, smooth $4-$manifold with intersection form $\Gamma$, $A(\Gamma)$ the automorphism group of $\Gamma$ and $D(M)$ the subgroup induced by orientation-preserving diffeomorphisms of $M$. In this note we study the…

The graph topology $\tau_{\Gamma}$ is the topology on the space $C(X)$ of all continuous functions defined on a Tychonoff space $X$ inherited from the Vietoris topology on $X\times \mathbb R$ after identifying continuous functions with…

It is known that if the special automorphism group $\text{SAut}(X)$ of a quasiaffine variety $X$ of dimension at least $2$ acts transitively on $X$, then this action is infinitely transitive. In this paper we address the question whether…

A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of…

An action of ${\mathbb Z}^k$ is associated to a higher rank graph $\Lambda$ satisfying a mild assumption. This generalises the construction of a topological Markov shift arising from a nonnegative integer matrix. We show that the stable…

We consider the dynamical system given by a diagonalizable element $a$ of a closed linear unimodular algebraic subgroup $G$ of the special linear group over the $p$-adic numbers acting by translation on a finite volume quotient $X$.…

We prove W$^*$-superrigidity for a large class of coinduced actions. We prove that if $\Sigma$ is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group $\Gamma$, the coinduced action…

We prove that for any compact zero-dimensional metric space $X$ on which an infinite countable amenable group $G$ acts freely by homeomorphisms, there exists a dynamical quasitiling with good covering, continuity, F{\o}lner and dynamical…

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $\Gamma$, there are uncountably many maximal subgroups of…

The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group \Gamma, with quotient group isomorphic to \Gamma/N. It is shown how to enumerate such…

Let ${\cal M}$ denote the Bose--Mesner algebra of a commutative $d$-class association scheme ${\mathfrak X}$ (not necessarily symmetric), and $\Gamma$ denote a (strongly) connected (directed) graph with adjacency matrix $A$. Under the…

A pair of graphs $(\Gamma,\Sigma)$ is called unstable if their direct product $\Gamma\times\Sigma$ admits automorphisms not from $\mathrm{Aut}(\Gamma)\times\mathrm{Aut}(\Sigma)$, and such automorphisms are said to be unexpected. The…

We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general…