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تسجيل مستخدم جديدTopological characteristic factors along cubes of minimal systems
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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 particular, for a distal minimal system, the maximal $(d-1)$-step pro-nilfactor is the topological cubic characteristic factor of order $d$.
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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
it closure $overline{mathcal O}((x, ldots,x), Ttimes T^2times ldots times T^d)$ is $pitimes ldots times pi$ ($d$ times) saturated. In 1994 Eli Glasner studied the topological characteristic factor for minimal systems. For example, it is shown that for a distal minimal system, its largest distal factor of order $d-1$ is its $d$-step topological characteristic factor. In this paper, we generalize Glasners work to the product system of finitely many minimal systems and give its relative version. To prove these results, we need to deal with $(X,T^m)$ for $min {mathbb {N}}$. We will study the structure theorem of $(X,T^m)$. We show that though for a minimal system $(X,T)$ and $min {mathbb {N}}$, $(X,T^m)$ may not be minimal, but we still can have PI-tower for $(X,T^m)$ and in fact it looks the same as the PI tower of $(X,T)$. We give some applications of the results developed. For example, we show that if a minimal system has no nontrivial independent pair along arithmetic progressions of order $d$, then up to a canonically defined proximal extension, it is PI of order $d$; if a minimal system $(X,T)$ has a nontrivial $d$-step topological characteristic factor, then there exist ``many $Delta$-transitive sets of order $d$.
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In this article we characterize measure theoretical eigenvalues of Toeplitz Bratteli-Vershik minimal systems of finite topological rank which are not associated to a continuous eigenfunction. Several examples are provided to illustrate the different situations that can occur.
Let $(X, T)$ be a weakly mixing minimal system, $p_1, cdots, p_d$ be integer-valued generalized polynomials and $(p_1,p_2,cdots,p_d)$ be non-degenerate. Then there exists a residual subset $X_0$ of $X$ such that for all $xin X_0$ $${ (T^{p_1(n)}x, cd
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