On Structure Theorems and Non-saturated Examples

Abstract

For any minimal system (X, T) and

d\ge 1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d\ge 1$$\end{document}

, there is an associated minimal system

(N_d(X),{\mathcal{G}}_d(T))

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(N_{d}(X), {\mathcal {G}}_{d}(T))$$\end{document}

, where

{\mathcal{G}}_d(T)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {G}}_{d}(T)$$\end{document}

is the group generated by

T\times \cdots \times T

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T\times \cdots \times T$$\end{document}

and

T\times T^2\times \cdots \times T^d

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T\times T^2\times \cdots \times T^{d}$$\end{document}

, and

N_d(X)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_{d}(X)$$\end{document}

is the orbit closure of the diagonal under

{\mathcal{G}}_d(T)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {G}}_{d}(T)$$\end{document}

. It is known that the maximal d-step pro-nilfactor of

N_d(X)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_d(X)$$\end{document}

is

N_d(X_d)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_d(X_d)$$\end{document}

, where

X_d

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_d$$\end{document}

is the maximal d-step pro-nilfactor of X. In this paper, we further study the structure of

N_d(X)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_d(X)$$\end{document}

. We show that the maximal distal factor of

N_d(X)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_d(X)$$\end{document}

is

N_d(X_{\mathit{dis}})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_d(X_{dis})$$\end{document}

with

X_{\mathit{dis}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_{dis}$$\end{document}

being the maximal distal factor of X, and prove that as minimal system

(N_d(X),{\mathcal{G}}_d(T))

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(N_{d}(X), {\mathcal {G}}_{d}(T))$$\end{document}

has the same structure theorem as (X, T). In addition, a non-saturated metric example (X, T) is constructed, which is not

T\times T^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T\times T^2$$\end{document}

-saturated and is a Toeplitz minimal system.

Keywords

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

References

[1] AkinE, GlasnerE, HuangW, ShaoS, YeX. Sufficient conditions under which a transitive system is chaotic. Ergodic Theory Dynam. Systems, 2010, 30: 1277-1310.

[2] Auslander, J.: Minimal flows and their extensions. North-Holland Mathematics Studies. 153, North-Holland, Amsterdam (1988)

[3] AuslanderJ, GlasnerE. The distal order of a minimal flow. Israel J. Math., 2002, 127: 61-80.

[4] CaiF, ShaoS. Topological characteristic factors along cubes of minimal systems. Discrete Contin. Dyn. Syst., 2019, 39: 5301-5317.

[5] Cai, F., Shao, S.: Topological characteristic factors and independence along arithmetic progressions. Israel J. Math., to appear. arXiv:2002.10963

[6] de VriesJElements of topological dynamics, 1993DordrechtKluwer Academic Publishers.

[7] DongP, DonosoS, MaassA, ShaoS, YeX. Infinite-step nilsystems, independence and complexity. Ergodic Theory Dynam. Systems, 2013, 33: 118-143.

[8] EllisR, GottschalkWH. Homomorphisms of transformation groups. Trans. Am. Math. Soc., 1960, 94: 272-281.

[9] EllisR, GlasnerS, ShapiroL. Proximal-isometric flows. Adv. Math., 1975, 17: 213-260.

[10] FollandGBA Course in Abstract Harmonic Analysis, 20162Baco RatonCRC Press.

[11] FurstenbergH. The structure of distal flows. Am. J. Math., 1963, 85: 477-515.

[12] GlasnerE. A topological ergodic decompositions and applications to products of a minimal transformation. J. Anal. Math., 1994, 64: 241-262.

[13] Glasner, E., Huang, W., Shao, S., Weiss, B., Ye, X.: Topological characteristic factors and nilsystems. J. Eur. Math. Soc., to appear. arXiv:2006.12385

[14] GlasnerE, GutmanY, YeX. Higher order regionally proximal equivalence relations for general group actions. Adv. Math., 2018, 333: 1004-1041.

[15] HostB, KraB. Nonconventional ergodic averages and nilmanifolds. Ann. Math. (2), 2005, 1611397-488.

[16] HostB, KraB, MaassA. Nilsequences and a structure theory for topological dynamical systems. Adv. Math., 2010, 224: 103-129.

[17] Lian, Z., Qiu, J.: Pro-nilfactors of the space of arithmetic progressions in topological dynamical systems. J. Dynam. Differential Equations, to appear. arXiv:2010.01803

[18] McMahonDC. Relativized weak disjointness and relatively invariant measures. Trans. Am. Math. Soc., 1978, 236: 225-237.

[19] ShaoS, YeX. Regionally proximal relation of order d\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d$$\end{document} is an equivalence one for minimal systems and a combinatorial consequence. Adv. Math., 2012, 231: 1786-1817.

[20] TuS, YeX. Dynamical parallelepipeds in minimal systems. J. Dynam. Differ. Equ., 2013, 25: 765-776.

[21] VeechWA. Point-distal flows. Am. J. Math., 1970, 92: 205-242.

[22] VeechWA. Topological dynamics. Bull. Am. Math. Soc., 1977, 83: 775-830.

[23] Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. ix+250 pp

[24] WilliamsS. Toeplitz minimal flows which are not uniquely ergodic. Z. Wahrsch. Verw. Gebiete, 1984, 67195-107.

[25] ZieglerT. Universal characteristic factors and Furstenberg averages. J. Am. Math. Soc., 2007, 20153-97.

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