Related papers: Strong universality, recurrence, and analytic P-id…

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…

Let $X$ be a partially ordered set with the property that each family of order intervals of the form $[a,b],[a,\rightarrow )$ with the finite intersection property has a nonempty intersection. We show that every directed subset of $X$ has a…

Let $X$ be a compact metric space and $f:X\to X$ a homeomorphism on $X$. We construct a fundamental domain for the set with finite peaks for each cocycle induced by $\phi\in C(X,R)$. In particular we prove that if a partially hyperbolic…

In 2019 Anthony Quas, Philippe Thieullen and Mohamed Zarrabi introduced the concept of strong fast invertibility for linear cocycles. It relates the growth of volumes between different initial times and, together with a condition on…

\def\G{\mathcal G} \def\M{\mathcal M} \def\cE{\mathcal E} We prove an analog of Liv\v{s}ic theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra $\G$ or a Lie group. Namely, we consider an…

An ideal $I$ is a family of subsets of positive integers $\textbf{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if for each…

The first goal of this article is to investigate a refinement of previously-introduced strongly regular hyperbolic automorphisms of locally finite thick Euclidean buildings $\Delta$ of finite Coxeter system $(W,S)$. The new ones are defined…

We show that within any strong orbit equivalent class, there exist minimal subshifts with arbitrarily low superlinear complexity. This is done by proving that for any simple dimension group with unit $(G,G^+,u)$ and any sequence of positive…

In this article, we prove that if a finitely generated group $G$ is not torsion then a necessary and sufficient condition for every full shift over $G$ has (continuous) cocycle superrigidity is that $G$ has one end. It is a topological…

We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are…

In this paper, we prove that if $T$ is diskcyclic operator then the closed unit disk multiplied by the union of the numerical range of all iterations of $T$ is dense in $\mathcal H$. Also, if $T$ is diskcyclic operator and $|\lambda|\le 1$,…

Many different definitions of computational universality for various types of dynamical systems have flourished since Turing's work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical…

We study the emergence of typicality in classical systems with a large number of binary state variables. We show analytically that for sufficiently large subsets of the complete state space, state functions which can be associated with…

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 present a reflexive Banach space $\mathfrak{X}_{_{^\text{usm}}}$ which is Hereditarily Indecomposable and satisfies the following properties. In every subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a weakly null normalized…

Given an ideal $\mathcal{I}$ on $\omega$, we prove that a sequence in a topological space $X$ is $\mathcal{I}$-convergent if and only if there exists a ``big'' $\mathcal{I}$-convergent subsequence. Then, we study several properties and show…

We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space X, which becomes a probability space…

In this paper, we will study the statistical behaviors of orbits. Firstly, we will show that for a dynamical systems have the shadowing property or almost specification property, the set of nonrecurrent points has full topological entropy.…

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal…

In this paper geometric properties of infinitely renormalizable real H\'enon-like maps $F$ in $\R^2$ are studied. It is shown that the appropriately defined renormalizations $R^n F$ converge exponentially to the one-dimensional…