Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts
Tomasz Downarowicz 1,, and Benjamin Weiss 2,
Faculty of Pure and Applied Mathematics, Wrocław University of Technology, Wrocław, Poland
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.
Keywords: Non-ergodic measure-preserving system, ergodic decomposition, pure topological model, strictly ergodic system, strictly uniform system.
Mathematics Subject Classification: Primary: 37B05, 37B20; Secondary: 37A25.
Citation: Tomasz Downarowicz, Benjamin Weiss. Pure strictly uniform models of non-ergodic measure automorphisms. Discrete & Continuous Dynamical Systems, 2022, 42 (2) : 863-884. doi: 10.3934/dcds.2021140
M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Sys., 3 (1983), 541-557. doi: 10.1017/S0143385700002133. Google Scholar
T. Downarowicz, Faces of simplexes of invariant measures, Israel J. Math., 165 (2008), 189-210. doi: 10.1007/s11856-008-1009-y. Google Scholar
T. Downarowicz and E. Glasner, Isomorphic extensions and applications, Topol. Meth. Nonlin. Analysis, 48 (2016), 321-338. doi: 10.12775/TMNA.2016.050. Google Scholar
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. Google Scholar
G. Hansel, Strict uniformity in ergodic theory, Math. Z., 135 (1974), 221-248. doi: 10.1007/BF01215027. Google Scholar
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. Google Scholar
R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech., 19 (1970), 717-729. Google Scholar
[8] A. S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995. doi: 10.1007/978-1-4612-4190-4. Google Scholar
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. Google Scholar
K. Kuratowski, Topology, Vol I, Academic press, New York, San Francisco, London, 1966. Google Scholar
E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239-255. doi: 10.1007/BF02772176. Google Scholar
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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Tomasz Downarowicz Benjamin Weiss