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Generic properties of extensions

Published online by Cambridge University Press:  13 March 2018

MIKE SCHNURR
MIKE SCHNURR*
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany email [email protected]
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Abstract

Motivated by the classical results by Halmos and Rokhlin on the genericity of weakly but not strongly mixing transformations and the Furstenberg tower construction, we show that weakly but not strongly mixing extensions on a fixed product space with both measures non-atomic are generic. In particular, a generic extension does not have an intermediate nilfactor.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 11 , November 2019 , pp. 3144 - 3168
Copyright
© Cambridge University Press, 2018 

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References

Ageev, O. N.. The generic automorphism of a Lebesgue space conjugate to a G-extension for any finite abelian group G. Dokl. Akad. Nauk 374 (2000), 439–442.Google Scholar
Ageev, O. N.. On the multiplicity function of generic group extensions with continuous spectrum. Ergod. Th. & Dynam. Sys. 21 (2001), 321–338.Google Scholar
Ageev, O. N.. On the genericity of some nonasymptotic dynamic properties. Uspekhi Mat. Nauk 58 (2003), 177–178.Google Scholar
Ageev, O. N.. The homogeneous spectrum problem in ergodic theory. Invent. Math. 160 (2005), 417–446.Google Scholar
Alpern, S. and Prasad, V. S.. Properties generic for Lebesgue space automorphisms are generic for measure-preserving manifold homeomorphisms. Ergod. Th. & Dynam. Sys. 22 (2002), 1587–1620.Google Scholar
Assani, I., Duncan, D. and Moore, R.. Pointwise characteristic factors for Wiener–Wintner double recurrence theorem. Ergod. Th. & Dynam. Sys. 36 (2016), 1037–1066.Google Scholar
Assani, I. and Presser, K.. Pointwise characteristic factors for the multiterm return times theorem. Ergod. Th. & Dynam. Sys. 32 (2012), 341–360.Google Scholar
Bergelson, V., Tao, T. and Ziegler, T.. Multiple recurrence and convergence results associated to F P 𝜔 -actions. J. Anal. Math. 127 (2015), 329–378.Google Scholar
Chacon, R. V.. Weakly mixing transformations which are not strongly mixing. Proc. Amer. Math. Soc. 22 (1969), 559–562.Google Scholar
Chu, Q.. Convergence of weighted polynomial multiple ergodic averages. Proc. Amer. Math. Soc. 137 (2009), 1363–1369.10.1090/S0002-9939-08-09614-7Google Scholar
Chu, Q., Frantzinakis, N. and Host, B.. Ergodic averages of commuting transformations with distinct degree polynomial iterates. Proc. Lond. Math. Soc. (3) 102 (2011), 801–842.Google Scholar
de la Rue, T. and de Sam Lazaro, J.. Une transformation générique peut être insérée dans un flot. Ann. Inst. Henri Poincaré Probab. Stat. 39 (2003), 121–134.Google Scholar
Einsiedler, M. and Ward, T.. Ergodic Theory with a View Towards Number Theory. Springer, New York, 2011.Google Scholar
Eisner, T. and Krause, B.. (Uniform) convergence of twisted ergodic averages. Ergod. Th. & Dynam. Sys. 36 (2016), 2172–2202.Google Scholar
Eisner, T. and Zorin-Kranich, P.. Uniformity in the Wiener–Wintner theorem for nilsequences. Discrete Contin. Dyn. Syst. 33 (2013), 3497–3516.Google Scholar
Frantzinakis, N. and Zorin-Kranich, P.. Multiple recurrence for non-commuting transformations along rationally independent polynomials. Ergod. Th. & Dynam. Sys. 35 (2015), 403–411.Google Scholar
Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerdi on arithmetic progressions. J. Anal. Math. 31 (1977), 204–256.Google Scholar
Furstenberg, H., Katznelson, Y. and Ornstein, D.. The ergodic theoretical proof of Szemeredi’s theorem. Bull. Amer. Math. Soc. (N.S.) 7(3) (1982), 527–552.Google Scholar
Glasner, E.. Ergodic Theory via Joinings. American Mathematical Society, Providence, RI, 2003.Google Scholar
Glasner, E. and Weiss, B.. Relative weak mixing is generic. Preprint, 2017, arXiv:1707.06425.Google Scholar
Guihneuf, P.-A.. Dynamical properties of spatial discretizations of a generic homeomorphism. Ergod. Th. & Dynam. Sys. 35 (2015), 1474–1523.Google Scholar
Halmos, P.. In general a measure preserving transformation is mixing. Ann. of Math. (2) 45(4) (1944), 786–792.Google Scholar
Halmos, P.. Lectures on Ergodic Theory. Chelsea Publishing Company, New York, 1956.Google Scholar
Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161 (2005), 397–488.Google Scholar
Host, B., Kra, B. and Maass, A.. Complexity of nilsystems and systems lacking nilfactors. J. Anal. Math. 124 (2014), 261–295.Google Scholar
Katok, A. B. and Stepin, A. M.. Metric properties of homeomorphisms that preserve measure. Uspekhi Mat. Nauk 25 (1970), 193–220.Google Scholar
King, J. L. F.. The generic transformation has roots of all orders. Colloq. Math. 84/85 (2000), 521–547.Google Scholar
Nadkarni, M. G.. Spectral theory of dynamical systems. Baire Category Theorems of Ergodic Theory. Birkhäuser, 1998, Ch. 8, pp. 51–61.Google Scholar
Robertson, D.. Characteristic factors for commuting actions of amenable groups. J. Anal. Math. 129 (2016), 165–196.Google Scholar
Robinson, E. A.. The maximal abelian subextension determines weak mixing for group extensions. Proc. Amer. Math. Soc. 114 (1992), 443–450.Google Scholar
Rokhlin, V.. A ‘general’ measure-preserving transformation is not mixing. Dokl. Akad. Nauk SSSR 60 (1948), 349–351.Google Scholar
Solecki, S.. Closed subgroups generated by generic measure automorphisms. Ergod. Th. & Dynam. Sys. 34 (2014), 1011–1017.Google Scholar
Tao, T.. Poincare’s legacies, Part II: pages from year two of a mathematical blog. Ergodic Theory. American Mathematical Society, Providence, RI, 2009, Ch. 2, pp. 161–355.Google Scholar
Tao, T. and Ziegler, T.. Concatenation theorems for anti-Gowers-uniform functions and Host–Kra characteristic factors. Discrete Anal. (2016).Google Scholar
Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20 (2007), 53–97.Google Scholar