Related papers: Cube structures of the universal minimal system, n…
In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In…
We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal $\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$. We study the structural properties of systems that…
We characterize inverse limits of nilsystems in topological dynamics, via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems. We provide two applications of the…
By proving the minimality of face transformations acting on the diagonal points and searching the points allowed in the minimal sets, it is shown that the regionally proximal relation of order $d$, $\RP^{[d]}$, is an equivalence relation…
We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $f \colon X\rightarrow Y$ between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of…
We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host-Kra seminorms. Our main result yields a complete structural description of cubic…
Kendall's Similarity Shape Theory for constellations of N points in the carrier space $\mathbb{R}^d$ as quotiented by the similarity group was developed for use in Probability and Statistics. It was subsequently shown to reside within…
We show that for a minimal system $(X,T)$, the set of saturated points along cubes with respect to its maximal $\infty$-step pro-nilfactor $X_\infty$ has a full measure. As an application, it is shown that if a minimal system $(X,T)$ has no…
This paper forms the first part of a series by the authors [GMV2,GMV3] concerning the structure theory of nilspaces of Antol\'in Camarena and Szegedy. A nilspace is a compact space $X$ together with closed collections of cubes…
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…
This paper forms the third part of a series by the authors [GMV1,GMV2] concerning the structure theory of nilspaces. A nilspace is a compact space $X$ together with closed collections of cubes $C^n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$,…
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 generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long standing problem of the existence of…
For any minimal system $(X,T)$ and $d\geq 1$ there is an associated minimal system $(N_{d}(X), \mathcal{G}_{d}(T))$, where $\mathcal{G}_{d}(T)$ is the group generated by $T\times\cdots\times T$ and $T\times T^2\times\cdots\times T^{d}$ and…
In this paper, we develop several structure theorems concerning commuting transformations and minimal $\mathbb{R}$-flows. Specifically, we show that if $(X,S)$, $(X,T)$ are minimal systems with $S$ and $T$ being commutative, then they…
A nilspace system is a generalization of a nilsystem, consisting of a compact nilspace X equipped with a group of nilspace translations acting on X. Nilspace systems appear in different guises in several recent works, and this motivates the…
Recent developments in ergodic theory, additive combinatorics, higher order Fourier analysis and number theory give a central role to a class of algebraic structures called nilmanifolds. In the present paper we continue a program started by…
Consider the semialgebraic structure over the real field. More generally, let an ominimal structure be over a real closed field. We show that a definable metric space X with a definable metric d is embedded into a Euclidean space so that…
We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This…
For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the regionally proximal relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We…