Pure strictly uniform models of non-ergodic measure automorphisms

Tomasz Downarowicz; Benjamin Weiss

doi:10.3934/dcds.2021140

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2022, Volume 42, Issue 2: 863-884. Doi: 10.3934/dcds.2021140

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Pure strictly uniform models of non-ergodic measure automorphisms

Tomasz Downarowicz^1, , and
Benjamin Weiss^2,

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
^* Corresponding author: Tomasz Downarowicz

Received: April 2021

Revised: August 2021

Early access: September 2021

Published: February 2022

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

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Abstract

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.

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

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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 $

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Figure 2. An array $ \hat x\in\hat{\mathfrak X} $

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Figure 3. Selected $ k $-rectangles from the array on Figure 2 (two $ 2 $-rectangles shaded dark-gray, and one $ 3 $-rectangle shaded light-gray)

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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)

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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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References

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