On Structure Theorems and Non-saturated Examples

Qinqi Wu , Hui Xu , Xiangdong Ye

Communications in Mathematics and Statistics ›› 2023, Vol. 13 ›› Issue (3) : 649 -685.

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Communications in Mathematics and Statistics ›› 2023, Vol. 13 ›› Issue (3) : 649 -685. DOI: 10.1007/s40304-022-00328-0
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On Structure Theorems and Non-saturated Examples

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Abstract

For any minimal system (XT) and

d1
, there is an associated minimal system
(Nd(X),Gd(T))
, where
Gd(T)
is the group generated by
T××T
and
T×T2××Td
, and
Nd(X)
is the orbit closure of the diagonal under
Gd(T)
. It is known that the maximal d-step pro-nilfactor of
Nd(X)
is
Nd(Xd)
, where
Xd
is the maximal d-step pro-nilfactor of X. In this paper, we further study the structure of
Nd(X)
. We show that the maximal distal factor of
Nd(X)
is
Nd(Xdis)
with
Xdis
being the maximal distal factor of X, and prove that as minimal system
(Nd(X),Gd(T))
has the same structure theorem as (XT). In addition, a non-saturated metric example (XT) is constructed, which is not
T×T2
-saturated and is a Toeplitz minimal system.

Keywords

Structure theorem / Minimal system / Saturation / Distal factor and pro-nilfactor

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Qinqi Wu, Hui Xu, Xiangdong Ye. On Structure Theorems and Non-saturated Examples. Communications in Mathematics and Statistics, 2023, 13(3): 649-685 DOI:10.1007/s40304-022-00328-0

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Funding

National Natural Science Foundation of China(12201599)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

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