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On structure theorems and non-saturated examples | | Qinqi Wu
; Hui Xu
; Xiangdong Ye
; | | Date: |
1 Jan 2022 | | Abstract: | For any minimal system $(X,T)$ and $dgeq 1$ there is an associated minimal
system $(N_{d}(X), mathcal{G}_{d}(T))$, where $mathcal{G}_{d}(T)$ is the
group generated by $T imescdots imes T$ and $T imes T^2 imescdots imes
T^{d}$ and $N_{d}(X)$ is the orbit closure of the diagonal under
$mathcal{G}_{d}(T)$. It is known that the maximal $d$-step pro-nilfactor of
$N_d(X)$ is $N_d(X_d)$, where $X_d$ is the maximal $d$-step pro-nilfactor of
$X$.
In this paper, we further study the structure of $N_d(X)$. We show that the
maximal distal factor of $N_d(X)$ is $N_d(X_{dis})$ with $X_{dis}$ being the
maximal distal factor of $X$, and prove that as minimal systems $(N_{d}(X),
mathcal{G}_{d}(T))$ has the same structure theorem as $(X,T)$. In addition, a
non-saturated metric example $(X,T)$ is constructed, which is not $T imes
T^2$-saturated and is a Toeplitz minimal system. | | Source: | arXiv, 2201.00152 | | Services: | Forum | Review | PDF | Favorites | |
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