Finite ergodic index and asymmetry for infinite measure preserving actions
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- by Alexandre I. Danilenko
- Proc. Amer. Math. Soc. 144 (2016), 2521-2532
- DOI: https://doi.org/10.1090/proc/12906
- Published electronically: October 5, 2015
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Abstract:
Given $k>0$ and an Abelian countable discrete group $G$ with elements of infinite order, we construct $(i)$ rigid funny rank-one infinite measure preserving (i.m.p.) $G$-actions of ergodic index $k$, $(ii)$ 0-type funny rank-one i.m.p. $G$-actions of ergodic index $k$, $(iii)$ funny rank-one i.m.p. $G$-actions $T$ of ergodic index 2 such that the product $T\times T^{-1}$ is not ergodic. It is shown that $T\times T^{-1}$ is conservative for each funny rank-one $G$-action $T$.References
- Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. MR 1450400, DOI 10.1090/surv/050
- T. M. Adams and C. E. Silva, On infinite transformations with maximal control of ergodic two-fold product powers, Isr. J. Math., to appear.
- J. Clancy, R. Friedberg, I. Kasmalkar, I. Loh, T. Pădurariu, C. E. Silva, and S. Vasudevan, Ergodicity and conservativity of products of infinite transformations and their inverses, arXiv:1408.2445.
- Alexandre I. Danilenko, $(C,F)$-actions in ergodic theory, Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser, Basel, 2008, pp. 325–351. MR 2402408, DOI 10.1007/978-3-7643-8608-5_{6}
- Alexandre I. Danilenko, Actions of finite rank: weak rational ergodicity and partial rigidity, Ergod. Th. & Dyn. Syst.
- Alexandre I. Danilenko and Valery V. Ryzhikov, On self-similarities of ergodic flows, Proc. Lond. Math. Soc. (3) 104 (2012), no. 3, 431–454. MR 2900232, DOI 10.1112/plms/pdr032
- A. I. Danilenko and C. E. Silva, Ergodic theory: Nonsingular transformations, Encyclopedia of Complexity and System Science, Springer 5 (2009), 3055–3083.
- S. Kakutani and W. Parry, Infinite measure preserving transformations with “mixing”, Bull. Amer. Math. Soc. 69 (1963), 752–756. MR 153815, DOI 10.1090/S0002-9904-1963-11022-8
- V. V. Ryzhikov, On the asymmetry of multiple asymptotic properties of ergodic actions, Math. Notes 96 (2014), no. 3-4, 416–422. Translation of Mat. Zametki 96 (2014), no. 3, 432–439. MR 3344315, DOI 10.1134/S0001434614090132
Bibliographic Information
- Alexandre I. Danilenko
- Affiliation: Institute for Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov, 61164, Ukraine
- MR Author ID: 265198
- Email: alexandre.danilenko@gmail.com
- Received by editor(s): December 13, 2014
- Received by editor(s) in revised form: June 19, 2015, and July 11, 2015
- Published electronically: October 5, 2015
- Communicated by: Nimish Shah
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2521-2532
- MSC (2010): Primary 37A40
- DOI: https://doi.org/10.1090/proc/12906
- MathSciNet review: 3477068