February  2022, 42(2): 863-884. doi: 10.3934/dcds.2021140

Pure strictly uniform models of non-ergodic measure automorphisms

1. 

Faculty of Pure and Applied Mathematics, Wrocław University of Technology, Wrocław, Poland

2. 

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

* Corresponding author: Tomasz Downarowicz

Received  April 2021 Revised  August 2021 Published  February 2022 Early access  September 2021

Fund Project: The first-named author is supported by National Science Center, Poland (Grant HARMONIA No. 2018/30/M/ST1/00061) and by the Wrocław University of Science and Technology

The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model.

Citation: Tomasz Downarowicz, Benjamin Weiss. Pure strictly uniform models of non-ergodic measure automorphisms. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 863-884. doi: 10.3934/dcds.2021140
References:
[1]

M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Sys., 3 (1983), 541-557.  doi: 10.1017/S0143385700002133.

[2]

T. Downarowicz, Faces of simplexes of invariant measures, Israel J. Math., 165 (2008), 189-210.  doi: 10.1007/s11856-008-1009-y.

[3]

T. Downarowicz and E. Glasner, Isomorphic extensions and applications, Topol. Meth. Nonlin. Analysis, 48 (2016), 321-338.  doi: 10.12775/TMNA.2016.050.

[4]

T. Downarowicz and B. Weiss, When all points are generic for ergodic measures, Bull. Polish Acad. Sci. Math., 68 (2020), 117-132.  doi: 10.4064/ba210113-15-1.

[5]

G. Hansel, Strict uniformity in ergodic theory, Math. Z., 135 (1974), 221-248.  doi: 10.1007/BF01215027.

[6]

B. Hasselblatt, Handbook of dynamical systems, Handbook of Dynamical Systems, Vol. 1A, 239–319, North-Holland, Amsterdam, (2002). doi: 10.1016/S1874-575X(02)80005-4.

[7]

R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech., 19 (1970), 717-729. 

[8] A. S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995.  doi: 10.1007/978-1-4612-4190-4.
[9]

W. Krieger, On unique ergodicity, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (1970), Vol. II, Berkeley-Los Angeles: University of California Press, (1972), 327–345.

[10]

K. Kuratowski, Topology, Vol I, Academic press, New York, San Francisco, London, 1966.

[11]

E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239-255.  doi: 10.1007/BF02772176.

show all references

References:
[1]

M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Sys., 3 (1983), 541-557.  doi: 10.1017/S0143385700002133.

[2]

T. Downarowicz, Faces of simplexes of invariant measures, Israel J. Math., 165 (2008), 189-210.  doi: 10.1007/s11856-008-1009-y.

[3]

T. Downarowicz and E. Glasner, Isomorphic extensions and applications, Topol. Meth. Nonlin. Analysis, 48 (2016), 321-338.  doi: 10.12775/TMNA.2016.050.

[4]

T. Downarowicz and B. Weiss, When all points are generic for ergodic measures, Bull. Polish Acad. Sci. Math., 68 (2020), 117-132.  doi: 10.4064/ba210113-15-1.

[5]

G. Hansel, Strict uniformity in ergodic theory, Math. Z., 135 (1974), 221-248.  doi: 10.1007/BF01215027.

[6]

B. Hasselblatt, Handbook of dynamical systems, Handbook of Dynamical Systems, Vol. 1A, 239–319, North-Holland, Amsterdam, (2002). doi: 10.1016/S1874-575X(02)80005-4.

[7]

R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech., 19 (1970), 717-729. 

[8] A. S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995.  doi: 10.1007/978-1-4612-4190-4.
[9]

W. Krieger, On unique ergodicity, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (1970), Vol. II, Berkeley-Los Angeles: University of California Press, (1972), 327–345.

[10]

K. Kuratowski, Topology, Vol I, Academic press, New York, San Francisco, London, 1966.

[11]

E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239-255.  doi: 10.1007/BF02772176.

Figure 1.  An array $ x\in{\mathfrak X} $. Each symbol $ x_{k,n} $ with $ k\ge 2 $ equals either $ 2x_{k-1,n} $ or $ 2x_{k-1,n}-1 $
Figure 2.  An array $ \hat x\in\hat{\mathfrak X} $
Figure 3.  Selected $ k $-rectangles from the array on Figure 2 (two $ 2 $-rectangles shaded dark-gray, and one $ 3 $-rectangle shaded light-gray)
Figure 4.  The top figure shows the classification of $ 2 $-rectangles into good and bad. The bottom figure shows bad rectangles replaced by the tabbed rectangles of the same size. Note that some markers in row 1 have moved (but this movement does not affect the construction)
Figure 5.  The tabbed rectangles $ R_l $ and $ \bar R_l $ are shown in the black frames. Their lengths are $ l $ and $ l+1 $, respectively. They start and end in the middle of copies of the base rectangle $ B_l $ (shown in gray)
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