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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 |
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.
References:
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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. |
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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. |
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H. Furstenberg,
The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.
doi: 10.2307/2373137. |
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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. |
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H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.
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E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/surv/101. |
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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. |
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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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