# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021140
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## 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 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021140
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##### References:
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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