Pure strictly uniform models of non-ergodic measure automorphisms

Tomasz Downarowicz; Benjamin Weiss

doi:10.3934/dcds.2021140

Article Contents

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

This issue Previous Article Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts Next Article Global stability solution of the 2D MHD equations with mixed partial dissipation

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: March 31, 2021

Revised: July 31, 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

Abstract Full Text(HTML) Figure(5) Related Papers Cited by

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.

Keywords:

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

Citation:

Full Text(HTML)

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 $

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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)

Download: Full-size image PowerPoint slide

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)

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.