Related papers: Saturation for product systems of polynomials

Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye, we prove that the maximal $\infty$-step pro-nilfactor $X_\infty$ of a minimal system $(X,T)$ is the topological characteristic factor along polynomials in a certain sense.…

We prove that the maximal infinite step pro-nilfactor $X_\infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $\pi:X…

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…

For a topological dynamical system $(X, T)$, $l\in\mathbb{N}$ and $x\in X$, let $N_l(X)$ and $L_x^l(X)$ be the orbit closures of the diagonal point $(x,x,\ldots,x)$ ($l $ times) under the actions $\mathcal{G}_{l}$ and $\tau_l $…

Let $\pi: (X,T)\rightarrow (Y,T)$ be a factor map of topological dynamics and $d\in {\mathbb {N}}$. $(Y,T)$ is said to be a $d$-step topological characteristic factor if there exists a dense $G_\delta$ set $X_0$ of $X$ such that for each…

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…

An $\infty$-step nilsystem is an inverse limit of minimal nilsystems. In this article is shown that a minimal distal system is an $\infty$-step nilsystem if and only if it has no nontrivial pairs with arbitrarily long finite IP-independence…

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…

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number $k$ of real continuous functions $f_1,\cdots, f_k$ such that the functions $f_i\circ T^n$, $n\in\mathbb Z$, $1\leq i\leq k,$ span a…

We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we…

A method how to construct Boolean-valued models of some fragments of arithmetic was developed in Krajicek (2011), with the intended applications in bounded arithmetic and proof complexity. Such a model is formed by a family of random…

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…

We prove that, for any type III$_1$ free product factor, its continuous core is full if and only if its $\tau$-invariant is the usual topology on the real line. This trivially implies, as a particular case, the same result for free…

For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems: systems for which the maximal measurable and maximal topological $k$-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such…

We give sufficient conditions on planar domains for polynomials to be dense in the algebras A and A-infinity of the product of these domains, endowed with their natural topologies. We also characterize the uniform limits, with respect to…

We show that for any topological dynamical system with approximate product property, the set of points whose forward orbits do not accumulate to any point in a large set carries full topological pressure.

In this paper we show that each polynomial exponential functor on complex finite-dimensional inner product spaces is defined up to equivalence of monoidal functors by an involutive solution to the Yang-Baxter equation (an involutive…

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the…

If $N \subset \R$ is a separable II$_1$-factor, the space $\Hom(N,\R)$ of unitary equivalence classes of unital *-homomorphisms $N \to \R$ is shown to have a surprisingly rich structure. If $N$ is not hyperfinite, $\Hom(N,\R)$ is an…

For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$…