# American Institute of Mathematical Sciences

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.

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##### 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.
An array $x\in{\mathfrak X}$. Each symbol $x_{k,n}$ with $k\ge 2$ equals either $2x_{k-1,n}$ or $2x_{k-1,n}-1$
An array $\hat x\in\hat{\mathfrak X}$
Selected $k$-rectangles from the array on Figure 2 (two $2$-rectangles shaded dark-gray, and one $3$-rectangle shaded light-gray)
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)
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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