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  2021, 17: 435-463. doi: 10.3934/jmd.2021015

A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system

Eli Glasner 1, and  Benjamin Weiss 2,

1. 

Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel
2. 

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Received  June 25, 2020 Revised  July 08, 2021 Published  November 2021

We show the existence, over an arbitrary infinite ergodic $ \mathbb{Z} $-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.

Keywords: Distal systems, structure theory, cocycles.
Mathematics Subject Classification: 37A05, 37A20.
Citation: Eli Glasner, Benjamin Weiss. A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system. Journal of Modern Dynamics, 2021, 17: 435-463. doi: 10.3934/jmd.2021015
References:
[1]

J. Aaronson and B. Weiss, On Herman's theorem for ergodic, amenable group extensions of endomorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 1283-1293.  doi: 10.1017/S0143385703000713.  Google Scholar

[2]

F. Beleznay and M. Foreman, The complexity of the collection of measure-distal transformations, Ergodic Theory Dynam. Systems, 16 (1996), 929-962.  doi: 10.1017/S0143385700010129.  Google Scholar

[3]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.  Google Scholar

[4]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar

[5] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.   Google Scholar
[6]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[7]

R. Jones and W. Parry, Compact abelian group extensions of dynamical systems II, Compositio Mathematica, 25 (1972), 135-147.   Google Scholar

[8]

A. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.  Google Scholar

[9]

W. Parry, Zero entropy of distal and related transformations, Topological Dynamics, (1968), 383–389.  Google Scholar

[10]

E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith., 27 (1975), 199-245.  doi: 10.4064/aa-27-1-199-245.  Google Scholar

[11]

B. Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 143-146.  doi: 10.1090/S0273-0979-1985-15399-6.  Google Scholar

[12]

R. J. Zimmer, Extensions of ergodic group actions, Illinois J. Math., 20 (1976), 373-409.  doi: 10.1215/ijm/1256049780.  Google Scholar

[13]

R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.  doi: 10.1215/ijm/1256049648.  Google Scholar

show all references

References:
[1]

J. Aaronson and B. Weiss, On Herman's theorem for ergodic, amenable group extensions of endomorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 1283-1293.  doi: 10.1017/S0143385703000713.  Google Scholar

[2]

F. Beleznay and M. Foreman, The complexity of the collection of measure-distal transformations, Ergodic Theory Dynam. Systems, 16 (1996), 929-962.  doi: 10.1017/S0143385700010129.  Google Scholar

[3]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.  Google Scholar

[4]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar

[5] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.   Google Scholar
[6]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[7]

R. Jones and W. Parry, Compact abelian group extensions of dynamical systems II, Compositio Mathematica, 25 (1972), 135-147.   Google Scholar

[8]

A. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.  Google Scholar

[9]

W. Parry, Zero entropy of distal and related transformations, Topological Dynamics, (1968), 383–389.  Google Scholar

[10]

E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith., 27 (1975), 199-245.  doi: 10.4064/aa-27-1-199-245.  Google Scholar

[11]

B. Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 143-146.  doi: 10.1090/S0273-0979-1985-15399-6.  Google Scholar

[12]

R. J. Zimmer, Extensions of ergodic group actions, Illinois J. Math., 20 (1976), 373-409.  doi: 10.1215/ijm/1256049780.  Google Scholar

[13]

R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.  doi: 10.1215/ijm/1256049648.  Google Scholar

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