Related papers: Regionally proximal relation of order d is an equi…

We introduce higher order regionally proximal relations suitable for an arbitrary acting group. For minimal abelian group actions, these relations coincide with the ones introduced by Host, Kra and Maass. Our main result is that these…

The regionally proximal relation of order $d$ along arithmetic progressions, namely ${\bf AP}^{[d]}$ for $d\in \N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${\bf AP}^{[d]}=\Delta$,…

We study several aspects of higher-order regionally proximal relations for group actions. First, we develop an algebraic approach to study higher-order regionally proximal relations. To this end, we introduce a new topology on a subgroup of…

We propose and develop an approach to study nilsystems and their proximal extensions using cube structures associated with the universal minimal system. We provide alternative proofs for results regarding saturation properties of factor…

We present a purely enveloping semigroup proof of a theorem of Shao and Ye which asserts that for an abelian group $T$, a minimal flow $(X,T)$ and any integer $d \ge 1$, the regional proximal relation of order $d$ is an equivalence…

Let $d\in {\mathbb N}$ and $p_i$ be an integral polynomial with $p_i(0)=0$, $1\le i\le d$. It is shown that if $S$ is piecewise syndetic in $\mathbb Z$, then $$\{(m,n)\in{\mathbb Z}^2: m+p_1(n),\ldots,m+p_d(n)\in S\}$$ is piecewise syndetic…

A classic family in topological dynamics is that of minimal rotations. One natural extension of this family is the class of nilsystems and their inverse limits. These systems have arisen in recent applications in ergodic theory and in…

In this paper, it is shown that for a minimal system $(X,T)$ and $d,k\in \mathbb{N}$, if $(x,x_i)$ is regionally proximal of order $d$ for $1\leq i\leq k$, then $(x,x_1,\ldots,x_k)$ is $(k+1)$-regionally proximal of order $d$. Meanwhile, we…

For an abelian group $G$, $\vec{g}=(g_1,\ldots,g_d)\in G^d$ and $\epsilon=(\epsilon(1),\ldots,\epsilon(d))\in \{0,1\}^d$, let $\vec{g}\cdot \epsilon=\prod_{i=1}^{d}g_i^{\epsilon(i)}$. In this paper, it is shown that for a minimal system…

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a…

In this paper, we introduce the notion of negatively regionally proximal pairs of onto maps which coincides with the set of regionally proximal pair of $f^{-1}$, whenever $f$ is an homeomorphism and we prove the maximal equicontinoues…

Downarowicz and Maass (2008) proposed topological ranks for all homeomorphic Cantor minimal dynamical systems using properly ordered Bratteli diagrams. In this study, we adopt this definition to the case of all essentially minimal…

In this paper, $d$-step almost automorphic systems are studied for $d\in\N$, which are the generalization of the classical almost automorphic ones. For a minimal topological dynamical system $(X,T)$ it is shown that the condition $x\in X$…

For a dynamical system $(X,T)$, $d\in\mathbb{N}$ and distinct non-constant integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, the notion of regionally proximal relation along $C=\{p_1,\ldots,p_d\}$ (denoted by $RP_C^{[d]}(X,T)$) is…

Any infinite sequence of substitutions with the same matrix of the Pisot type defines a symbolic dynamical system which is minimal. We prove that, to any such sequence, we can associate a compact set (Rauzy fractal) by projection of the…

The starting point for this work is an identity that relates the number of minimal matrices with prescribed 1-marginals and coefficient sequence to a linear combination of Kronecker coefficients. In this paper we provide a bijection that…

We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being `small' in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation…

This paper is a systematic study about the syndetically proximal relation and the possible existence of syndetically scrambled sets for the dynamics of continuous self-maps of compact metric spaces. Especially we consider various classes of…

A topological setting is defined to study the complexities of the relation of equivalence of embeddings (or "position") of a Banach space into another and of the relation of isomorphism of complex structures on a real Banach space. The…

In this paper we characterize the definiteness of the discrete symplectic system, study a nonhomogeneous discrete symplectic system, and introduce the minimal and maximal linear relations associated with these systems. Fundamental…