Strictly Ergodic Models Under Face and Parallelepiped Group Actions

Abstract

The Jewett–Krieger theorem states that each ergodic system has a strictly ergodic topological model. In this article, we show that for an ergodic system one may require more properties on its strictly ergodic model. For example, the orbit closure of points in diagonal under face transforms may be also strictly ergodic. As an application, we show the pointwise convergence of ergodic averages along cubes, which was firstly proved by Assani (J Anal Math 110:241–269, 2010).

Keywords

Ergodic averages / Model / Cubes / Face transformations

References

[1] Assani I. Pointwise convergence of ergodic averages along cubes. J. Anal. Math.. 2010, 110 241-269

[2] Bergelson V. The Multifarious Poincare Recurrence Theorem, Descriptive Set Theory and Dynamical Systems, London Mathematical Society Lecture Note Series. 2000 Cambridge: Cambridge University Press

[3] Bergelson, V.: Combinatorial and Diophantine applications of ergodic theory, Appendix A by A. Leibman and Appendix B by Anthony Quas and Máté Wierdl. In: Hasselblatt, B., Katok, A. (eds.) Handbook of Dynamical Systems. vol. 1B, pp. 745–869. Elsevier B. V., Amsterdam (2006)

[4] Chu Q, Frantzikinakis N. Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations. Ergod. Theory Dyn. Syst.. 2012, 32 877-897

[5] Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. M. B. Porter Lectures. 1981 Princeton: Princeton University Press

[6] Furstenberg H, Weiss B. On almost $1$-$1$ extensions. Israel J. Math.. 1989, 65 3 311-322

[7] Glasner E. Ergodic Theory via Joinings, Mathematical Surveys and Monographs. 2003 Providence: American Mathematical Society

[8] Glasner E, Weiss B. Strictly ergodic, uniform positive entropy models. Bull. Soc. Math. France. 1994, 122 3 399-412

[9] Host B, Kra B. Nonconventional averages and nilmanifolds. Ann. Math.. 2005, 161 398-488

[10] Host B, Kra B. Uniformity norms on $l^\infty $ and applications. J. Anal. Math.. 2009, 108 219-276

[11] Host B, Kra B, Maass A. Nilsequences and a structure theory for topological dynamical systems. Adv. Math.. 2010, 224 103-129

[12] Huang, W., Shao, S., Ye, X.D.: Nil Bohr-Sets and Almost Automorphy of Higher Order, Mem. Amer. Math. Soc. 241(1143) (2016)

[13] Huang, W., Shao, S., Ye, X.: Pointwise convergence of multiple ergodic averages and strictly ergodic models, preprint. arXiv:1406.5930

[14] Jewett, R.I.: The prevalence of uniquely ergodic systems. J. Math. Mech. 19, 717–729 (1969/1970)

[15] Krieger, W.: On unique ergodicity. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), vol. II: Probability Theory, pp. 327–346. University California Press, Berkeley, California (1972)

[16] Lehrer E. Topological mixing and uniquely ergodic systems. Israel J. Math.. 1987, 57 2 239-255

[17] Lindenstrauss E. Pointwise theorems for amenable groups. Invent. Math.. 2001, 146 259-295

[18] Shao S, Ye X. Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence. Adv. Math.. 2012, 231 1786-1817

[19] Weiss B. Strictly ergodic models for dynamical systems. Bull. Am. Math. Soc. (N.S.). 1985, 13 143-146

[20] Weiss B. Countable generators in dynamics—universal minimal models. Contemp. Math.. 1989, 94 321-326

[21] Ziegler T. Universal characteristic factors and Furstenberg averages. J. Am. Math. Soc.. 2007, 20 1 53-97

Funding

NNSF(11225105)

NNSF of China(11371339)

NNSF of China(11571335)

the Fundamental Research Funds for the Central Universities