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In this paper, it is shown that for $dinmathbb{N}$, a minimal system $(X,T)$ is a $d$-step pro-nilsystem if its enveloping semigroup is a $d$-step top-nilpotent group, answering an open question by Donoso. Thus, combining the previous result of Donoso, it turns out that a minimal system $(X,T)$ is a $d$-step pro-nilsystem if and only if its enveloping semigroup is a $d$-step top-nilpotent group.
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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 $pi:X rightarrow X_infty$, the induced open extension $pi^*:X^* rightarrow X^*_infty$ has the following property: for $x$ in a dense $G_delta$ set of $X^*$, the orbit closure $L_x=overline{{mathcal{O}}}((x,x,ldots,x), Ttimes T^2times ldots times T^d)$ is $(pi^*)^{(d)}$-saturated, i.e. $L_x=((pi^*)^{(d)})^{-1}(pi^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $kge 2$, then for any $din {mathbb N}$ and any $0le j<k$ there is a sequence ${n_i}$ of $mathbb Z$ with $n_iequiv j (text{mod} k)$ such that $T^{n_i}xrightarrow x, T^{2n_i}xrightarrow x, ldots, T^{dn_i}xrightarrow x$ for $x$ in a dense $G_delta$ subset of $X$; (2) if $(X,T)$ is totally minimal, then ${T^{n^2}x:nin {mathbb Z}}$ is dense in $X$ for $x$ in a dense $G_delta$ subset of $X$; (3) for any $dinmathbb N$ and any minimal system, which is an open extension of its maximal distal factor, ${bf RP}^{[d]}={bf AP}^{[d]}$, where the latter is the regionally proximal relation of order $d$ along arithmetic progressions.
In this paper, we study affine transformations on tori, nilmanifolds and compact abelian groups. For these systems, we show that an equivalent condition for zero entropy is the orbit closure of each point has a nice structure. To be precise, the affine systems on those spaces are zero entropy if and only if the orbit closure of each point is isomorphic to an inverse limit of nilsystems.
The authors have shown previously that every locally pro-p contraction group decomposes into the direct product of a p-adic analytic factor and a torsion factor. It has long been known that p-adic analytic contraction groups are nilpotent. We show here that the torsion factor is nilpotent too, and hence that every locally pro-p contraction group is nilpotent.
We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field.
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 ergodic measure of $(X,T)$ is isomorphic to a $d$-step pro-nilsystem, and thus $(X,T)$ has zero entropy. Moreover, it is shown that if $(X,T)$ is a strictly ergodic distal system with the property that the maximal topological and measurable $d$-step pro-nilsystems are isomorphic, then ${bf AP}^{[d]}={bf RP}^{[d]}$ for each $din {mathbb N}$. It follows that for a minimal $infty$-pro-nilsystem, ${bf AP}^{[d]}={bf RP}^{[d]}$ for each $din {mathbb N}$. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.
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