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

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

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.

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.

[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.

[3]

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

[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.

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

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

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

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.

[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.

[3]

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

[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.

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

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

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

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