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# Pure strictly uniform models of non-ergodic measure automorphisms

• * Corresponding author: Tomasz Downarowicz

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.

Mathematics Subject Classification: Primary: 37B05, 37B20; Secondary: 37A25.

 Citation: • • 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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