ﻻ يوجد ملخص باللغة العربية
For a topological dynamical system $(X, T)$, $linmathbb{N}$ and $xin X$, 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 $ respectively, where $mathcal{G}_{l}$ is generated by $Ttimes Ttimes ldots times T$ ($l $ times) and $tau_l=Ttimes T^2times ldots times T^l$. In this paper, we show that for a minimal system $(X,T)$ and $lin mathbb{N}$, the maximal $d$-step pro-nilfactor of $(N_l(X),mathcal{G}_{l})$ is $(N_l(X_d),mathcal{G}_{l})$, where $pi_d:Xto X/mathbf{RP}^{[d]}=X_d,din mathbb{N}$ is the factor map and $mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$. Meanwhile, when $(X,T)$ is a minimal nilsystem, we also calculate the pro-nilfactors of $(L_x^l(X),tau_l)$ for almost every $x$ w.r.t. the Haar measure. In particular, there exists a minimal $2$-step nilsystem $(Y,T)$ and a countable set $Omegasubset Y$ such that for $yin Ybackslash Omega$ the maximal equicontinuous factor of $(L_y^2(Y),tau_2)$ is not $(L_{pi_1(y)}^2(Y_{1}),tau_2)$.
Let $pi: (X,T)rightarrow (Y,T)$ be a factor map of topological dynamics and $din {mathbb {N}}$. $(Y,T)$ 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 $xin X_0$ the orb
The regionally proximal relation of order $d$ along arithmetic progressions, namely ${bf AP}^{[d]}$ for $din N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${bf AP}^{[d]}=Delta$, then each ergo
We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an unlikely intersection statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dyna
Let $mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,{f_n}_{n=1}^{+infty})$
We introduce the notion of Bohr chaoticity, which is a topological invariant for topological dynamical systems, and which is opposite to the property required by Sarnaks conjecture. We prove the Bohr chaoticity for all systems which have a horseshoe