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Actions of finite rank: weak rational ergodicity and partial rigidity

Published online by Cambridge University Press:  13 April 2015

ALEXANDRE I. DANILENKO
ALEXANDRE I. DANILENKO*
Affiliation:
Institute for Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov, 61164, Ukraine email alexandre.danilenko@gmail.com
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Abstract

A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence weakly rationally ergodic. A constructive definition of finite funny rank for actions of arbitrary infinite countable groups is given. It is shown that the ergodic i.m.p. transformations of balanced finite funny rank are subsequence weakly rationally ergodic. It is shown that the ergodic probability preserving transformations of exact finite rank, the ergodic Bratteli–Vershik maps corresponding to the ‘consecutively ordered’ Bratteli diagrams of finite rank, some their generalizations and the ergodic interval exchange transformations are partially rigid.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2138 - 2171
Copyright
© Cambridge University Press, 2015 

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References

Aaronson, J.. Rational ergodicity and a metric invariant for Markov shifts. Israel J. Math. 27 (1977), 93–123.CrossRefGoogle Scholar
Aaronson, J.. Rational weak mixing in infinite measure spaces. Ergod. Th. & Dynam. Sys. 33 (2013), 1611–1643.CrossRefGoogle Scholar
Aaronson, J.. Conditions for rational weak mixing. Preprint, 2012, arXiv:1208.3505.Google Scholar
Bezuglyi, S. I. and Golodets, V. Ya.. Hyperfinite and II 1 actions for nonamenable groups. J. Funct. Anal. 40 (1981), 30–44.CrossRefGoogle Scholar
Bezuglyi, S., Kwiatkowski, J., Medynets, K. and Solomyak, B.. Finite rank Bratteli diagrams: structure of invariant measures. Trans. Amer. Math. Soc. 365 (2013), 2637–2679.CrossRefGoogle Scholar
Bozgan, F., Sanchez, A., Silva, C. E., Stevens, D. and Wang, J.. Subsequence rational ergodicity of rank-one transformations. Preprints, 2013, arXiv:1310.5084v1 and arXiv:1310.5084v2.CrossRefGoogle Scholar
Dai, I., Garcia, X., Padurariu, T. and Silva, C. E.. On rationally ergodic and rationally weakly mixing rank-one transformations. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
Danilenko, A. I.. Funny rank one weak mixing for nonsingular Abelian actions. Israel J. Math. 121 (2001), 29–54.CrossRefGoogle Scholar
Danilenko, A. I.. Strong orbit equivalence of locally compact Cantor minimal systems. Internat. J. Math. 12 (2001), 113–123.CrossRefGoogle Scholar
Danilenko, A. I.. (C, F)-actions in ergodic theory. Geometry and Dynamics of Groups and Spaces (Progress in Mathematics, 265) . Birkhäuser, Basel, 2008, pp. 325–351.CrossRefGoogle Scholar
Danilenko, A. I. and Silva, C. E.. Ergodic theory: nonsingular transformations. Encyclopedia of Complexity and Systems Science. Springer, New York, 2009, pp. 3055–3083.CrossRefGoogle Scholar
del Junco, A.. A simple map with no prime factors. Israel J. Math. 104 (1998), 301–320.CrossRefGoogle Scholar
Durand, F.. Combinatorics on Bratteli diagrams and dynamical systems. Combinatorics, Automata and Number Theory (Encyclopedia of Mathematics and its Applications, 135) . Eds. Berthé, V. and Rigo, M.. Cambridge University Press, Cambridge, 2010, pp. 338–386.Google Scholar
Ferenczi, S.. Systèmes de rang un gauche. Ann. Inst. H. Poincaré Probab. Statist. 21 (1985), 177–186.Google Scholar
Ferenczi, S.. Systems of finite rank. Colloq. Math. 73 (1997), 35–65.CrossRefGoogle Scholar
Gjerde, R. and Johansen, Ø.. Bratteli–Vershik models for Cantor minimal systems associated to interval exchange transformations. Math. Scand. 90 (2002), 87–100.CrossRefGoogle Scholar
Katok, A.. Interval exchange transformations and some special flows are not mixing. Israel J. Math. 35 (1980), 301–310.CrossRefGoogle Scholar
Kornfel’d, I. P., Sinai, Ya. G. and Fomin, S. V.. Ergodic Theory. Nauka, Moscow, 1980 (in Russian).Google Scholar
Matui, H.. Topological orbit equivalence of locally compact Cantor minimal systems. Ergod. Th. & Dynam. Sys. 22 (2002), 1871–1903.CrossRefGoogle Scholar
Rosenthal, A.. Les systèmes de rang fini exact ne sont pas mélangeants. Preprint, 1984.Google Scholar
Ryzhikov, V. V.. The absence of mixing in special flows over rearrangements of segments. Math. Notes 55 (1994), 648–650.CrossRefGoogle Scholar
Weiss, B.. Strictly ergodic models for dynamical systems. Bull. Amer. Math. Soc. (N.S.) 13 (1985), 143–146.CrossRefGoogle Scholar
Weiss, B.. Monotileable amenable groups. Topology, Ergodic Theory, Real Algebraic Geometry (American Mathematical Society Translation Series 2, 202) . American Mathematical Society, Providence, RI, 2001, pp. 257–262.Google Scholar
Yuasa, H.. Invariant measures for the subshifts arising from non-primitive substitutions. J. Anal. Math. 102 (2007), 143–180.CrossRefGoogle Scholar
Yuasa, H.. Uniform sets for infinite measure-preserving systems. J. Anal. Math. 120 (2013), 333–356.CrossRefGoogle Scholar
Zimmer, R.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Anal. 27 (1978), 350–372.CrossRefGoogle Scholar